Christopher Dougherty EC220 - Introduction to econometrics (chapter 4) Slideshow: comparing linear and logarithmic specifications Original citation: Dougherty, C.
Download ReportTranscript Christopher Dougherty EC220 - Introduction to econometrics (chapter 4) Slideshow: comparing linear and logarithmic specifications Original citation: Dougherty, C.
Slide 1
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 2
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 3
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 4
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 5
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 6
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 7
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 8
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 9
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 10
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 11
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 12
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 13
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 14
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 15
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 16
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 17
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 18
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 19
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 20
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 21
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 22
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 23
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 24
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 25
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 26
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 27
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 28
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 29
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 30
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 2
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 3
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 4
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 5
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 6
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 7
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 8
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 9
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 10
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 11
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 12
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 13
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 14
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 15
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 16
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 17
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 18
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 19
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 20
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 21
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 22
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 23
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 24
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 25
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 26
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 27
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 28
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 29
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
Slide 30
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: comparing linear and logarithmic specifications
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/130/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
When alternative specifications of a regression model have the same dependent variable, R2
can be used to compare their goodness of fit.
1
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
However, when the dependent variable is different, this is not legitimate.
2
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
In the case of the linear model, R2 measures the proportion of the variance in Y explained by
the model. In the case of the semilogarithmic model, it measures the proportion of the
variance of the logarithm of Y explained by the model.
3
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Clearly these are related, but they are not the same and direct comparisons are not valid.
4
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
However, the goodness of fit of models with linear and logarithmic versions of the same
dependent variable can be compared indirectly by subjecting the dependent variable to the
Box–Cox transformation and fitting the model shown.
5
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
This is a family of specifications that depend on the parameter . The determination of is
an empirical matter, like the determination of the other parameters.
6
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
1 2 X u
The model is nonlinear in parameters and so a nonlinear regression method should be
used. In practice, maximum likelihood estimation is used.
7
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The reason that this transformation is of interest in the present context is that
specifications with linear and logarithmic dependent variables are special cases.
8
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 1 gives the linear model. The dependent variable is then Y – 1, rather than Y, but
subtracting a constant from the dependent variable does not affect the regression results,
except for the estimate of the intercept.
9
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
Putting = 0 gives the (semi–)logarithmic model. Of course, one cannot talk about putting
exactly equal to 0, because then the dependent variable becomes zero divided by zero. We
are talking about the limiting form as tends to zero.
10
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
So one could fit the general model and see whether is close to 0 or close to 1. Of course.
'close' has no meaning in econometrics. To approach this issue technically, one should test
the hypotheses = 0 and = 1.
11
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
The outcome might be that one is rejected and the other not rejected, but of course it is
possible that neither might be rejected or both might be rejected, given your chosen
significance level.
12
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Box–Cox transformation:
Y 1
Y 1
Y 1
1 2 X u
Y 1
when
1
log Y
when
0
If you are interested only in comparing the fits of the linear and logarithmic specifications,
there is a short-cut procedure that involves only standard least squares regressions.
13
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
The first step is to divide the observations on the dependent variable by their geometric
mean. We will call the transformed variable Y*.
14
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
You now regress Y* and logeY*, leaving the right side of the equation unchanged. (The
parameters have been given prime marks to emphasize that the coefficients will not be
estimates of the original 1 and 2.)
15
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
The residual sums of squares are now directly comparable. The specification with the
smaller RSS therefore provides the better fit.
16
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
Y 1 2 X u
logY 1 2 X u
Y* Y /
geometric mean of Y
Y * 1' 2' X u
log Y * 1' 2' X u
We will use the transformation to compare the fits of the linear and semilogarithmic
versions of a simple earnings function, using EAEF Data Set 21.
17
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The first step is to calculate the geometric mean of the dependent variable. The easiest way
to do this is to take the exponential of the mean of the log of the dependent variable.
18
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
The sum of the logarithms of Y is equal to the logarithm of the products of Y.
19
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
Now we use the rule that alog X is the same as log Xa.
20
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
And finally we use the fact that the exponential of the logarithm of X reduces to X.
21
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.79)
. gen LGEARNST = ln(EARNSTAR)
LGEARN has already been defined as the logarithm of EARNINGS. We find its mean. In
Stata this is done with the ‘sum’ command.
22
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We then define EARNSTAR, dividing EARNINGS by the exponential of the mean of LGEARN.
23
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
e
1
n
logYi
e
e
1
log(Y1Y2 ...Yn )
n
log(Y1Y2 ...Yn
1
)n
(Y1Y2 ...Yn )
1
n
. sum LGEARN
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------LGEARN |
540
2.791993
.5885545
.7561221
4.789074
. gen EARNSTAR = EARNINGS/exp(2.7920)
. gen LGEARNST = ln(EARNSTAR)
We also define LGEARNST, the logarithm of EARNSTAR.
24
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg EARNSTAR S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 84.5951503
2 42.2975751
Residual | 336.283886
537 .626226977
-------------+-----------------------------Total | 420.879037
539 .780851645
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
.79135
-----------------------------------------------------------------------------EARNSTAR |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.164165
.0143224
11.46
0.000
.1360303
.1922998
EXP |
.0344763
.0078777
4.38
0.000
.0190014
.0499511
_cons | -1.623491
.2618984
-6.20
0.000
-2.137962 -1.10902
------------------------------------------------------------------------------
Here is the regression of EARNSTAR on S and EXP. The residual sum of squares is 336.3.
25
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. reg LGEARNST S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 50.9842592
2 25.4921296
Residual | 135.723388
537 .252743739
-------------+-----------------------------Total | 186.707647
539 .346396377
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
100.86
0.0000
0.2731
0.2704
.50274
-----------------------------------------------------------------------------LGEARNST |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
.1235911
.0090989
13.58
0.000
.1057173
.141465
EXP |
.0350826
.0050046
7.01
0.000
.0252515
.0449137
_cons |
-2.28268
.1663823
-13.72
0.000
-2.60952
-1.95584
------------------------------------------------------------------------------
We run the parallel regression for LGEARNST. The residual sum of squares is 135.7. Thus
we conclude that the semilogarithmic version gives a better fit.
26
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
Here is the output for the full Box–Cox regression. The parameter that we have denoted
(lambda) is called theta by Stata. It is estimated at –0.13. Since it is closer to 0 than to 1, it
indicates that the dependent variable should be logarithmic rather than linear.
27
COMPARING LINEAR AND LOGARITHMIC SPECIFICATIONS
. boxcox EARNINGS S EXP
Log likelihood = -1897.7671
Number of obs
LR chi2(2)
Prob > chi2
=
=
=
540
172.45
0.000
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/theta |
-.129785
.0506665
-2.56
0.010
-.2290894
-.0304805
-------------------------------------------------------------------------------------------------------------------------------------Test
Restricted
LR statistic
P-value
H0:
log likelihood
chi2
Prob > chi2
--------------------------------------------------------theta = -1
-2048.3423
301.15
0.000
theta = 0
-1901.0466
6.56
0.010
theta = 1
-2146.0312
496.53
0.000
---------------------------------------------------------
However, even the value of 0 does not lie in the 95 percent confidence interval. (The log
likelihood tests will be explained in Chapter 10.)
28
Copyright Christopher Dougherty 2011.
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The content of this slideshow comes from Section 4.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
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11.07.25