Christopher Dougherty EC220 - Introduction to econometrics (chapter 4) Slideshow: elasticities and logarithmic models Original citation: Dougherty, C.
Download ReportTranscript Christopher Dougherty EC220 - Introduction to econometrics (chapter 4) Slideshow: elasticities and logarithmic models Original citation: Dougherty, C.
Slide 1
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 4)
Slideshow: elasticities and logarithmic models
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/
Slide 2
ELASTICITIES AND LOGARITHMIC MODELS
Elasticity of Y with respect to
X
is the proportional change in Y
per proportional change in X:
elasticity
dY Y
dX X
A
Y
dY dX
Y X
Y
X
O0
X
52
This sequence defines elasticities and shows how one may fit nonlinear models with
constant elasticities. First, the general definition of an elasticity.
1
Slide 3
ELASTICITIES AND LOGARITHMIC MODELS
Elasticity of Y with respect to
X
is the proportional change in Y
per proportional change in X:
elasticity
dY Y
dX X
A
Y
dY dX
Y X
slope of the tangent at A
slope of OA
Y
X
O0
X
52
Re-arranging the expression for the elasticity, we can obtain a graphical interpretation.
2
Slide 4
ELASTICITIES AND LOGARITHMIC MODELS
Elasticity of Y with respect to
X
is the proportional change in Y
per proportional change in X:
elasticity
dY Y
dX X
A
Y
dY dX
Y X
slope of the tangent at A
slope of OA
Y
X
O0
X
52
The elasticity at any point on the curve is the ratio of the slope of the tangent at that point to
the slope of the line joining the point to the origin.
3
Slide 5
ELASTICITIES AND LOGARITHMIC MODELS
Elasticity of Y with respect to
X
is the proportional change in Y
per proportional change in X:
elasticity
dY Y
dX X
A
Y
dY dX
Y X
slope of the tangent at A
slope of OA
Y
X
O0
X
52
elasticity < 1
In this case it is clear that the tangent at A is flatter than the line OA and so the elasticity
must be less than 1.
4
Slide 6
ELASTICITIES AND LOGARITHMIC MODELS
Elasticity of Y with respect to
X
is the proportional change in Y
per proportional change in X:
elasticity
dY Y
dX X
A
Y
dY dX
Y X
slope of the tangent at A
slope of OA
O0
X
52
elasticity > 1
In this case the tangent at A is steeper than OA and the elasticity is greater than 1.
5
Slide 7
ELASTICITIES AND LOGARITHMIC MODELS
Y 1 2X
elasticity
Y
A
dY dX
Y X
slope of the tangent at A
slope of OA
2
(1 2 X ) / X
O
Xx
2
(1 / X ) 2
In general the elasticity will be different at different points on the function relating Y to X.
6
Slide 8
ELASTICITIES AND LOGARITHMIC MODELS
Y 1 2X
elasticity
Y
A
dY dX
Y X
slope of the tangent at A
slope of OA
2
(1 2 X ) / X
O
Xx
2
(1 / X ) 2
In the example above, Y is a linear function of X.
7
Slide 9
ELASTICITIES AND LOGARITHMIC MODELS
Y 1 2X
elasticity
Y
A
dY dX
Y X
slope of the tangent at A
slope of OA
2
(1 2 X ) / X
O
Xx
2
(1 / X ) 2
The tangent at any point is coincidental with the line itself, so in this case its slope is always
2. The elasticity depends on the slope of the line joining the point to the origin.
8
Slide 10
ELASTICITIES AND LOGARITHMIC MODELS
Y 1 2X
Y
B
elasticity
A
dY dX
Y X
slope of the tangent at A
slope of OA
2
(1 2 X ) / X
O
Xx
2
(1 / X ) 2
OB is flatter than OA, so the elasticity is greater at B than at A. (This ties in with the
mathematical expression: (1 / X) + 2 is smaller at B than at A, assuming that 1 is positive.)
9
Slide 11
ELASTICITIES AND LOGARITHMIC MODELS
Y 1X
2
However, a function of the type shown above has the same elasticity for all values of X.
10
Slide 12
ELASTICITIES AND LOGARITHMIC MODELS
Y 1X
dY
dX
2
1 2 X
2 1
For the numerator of the elasticity expression, we need the derivative of Y with respect to X.
11
Slide 13
ELASTICITIES AND LOGARITHMIC MODELS
Y 1X
dY
dX
Y
X
2
1 2 X
1X
X
2
2 1
1X
2 1
For the denominator, we need Y/X.
12
Slide 14
ELASTICITIES AND LOGARITHMIC MODELS
Y 1X
dY
dX
Y
X
elasticity
2
1 2 X
1X
2
X
dY dX
Y X
2 1
1X
2 1
1 2 X
1X
2 1
2 1
2
Hence we obtain the expression for the elasticity. This simplifies to 2 and is therefore
constant.
13
Slide 15
ELASTICITIES AND LOGARITHMIC MODELS
Y
Y 1X
2
2 0 . 25
X
By way of illustration, the function will be plotted for a range of values of 2. We will start
with a very low value, 0.25.
14
Slide 16
ELASTICITIES AND LOGARITHMIC MODELS
Y
Y 1X
2
2 0 . 50
X
We will increase 2 in steps of 0.25 and see how the shape of the function changes.
15
Slide 17
ELASTICITIES AND LOGARITHMIC MODELS
Y
Y 1X
2
2 0 . 75
X
16
Slide 18
ELASTICITIES AND LOGARITHMIC MODELS
Y
Y 1X
2
2 1 . 00
X
When 2 is equal to 1, the curve becomes a straight line through the origin.
17
Slide 19
ELASTICITIES AND LOGARITHMIC MODELS
Y
Y 1X
2
2 1 . 25
X
18
Slide 20
ELASTICITIES AND LOGARITHMIC MODELS
Y
Y 1X
2
2 1 . 50
X
19
Slide 21
ELASTICITIES AND LOGARITHMIC MODELS
Y
Y 1X
2
2 1 . 75
X
Note that the curvature can be quite gentle over wide ranges of X.
20
Slide 22
ELASTICITIES AND LOGARITHMIC MODELS
Y
Y 1X
2
2 2 . 00
X
This means that even if the true model is of the constant elasticity form, a linear model may
be a good approximation over a limited range.
21
Slide 23
ELASTICITIES AND LOGARITHMIC MODELS
Y 1X
2
log Y log 1 X
2
log 1 log X
2
log 1 2 log X
It is easy to fit a constant elasticity function using a sample of observations. You can
linearize the model by taking the logarithms of both sides.
22
Slide 24
ELASTICITIES AND LOGARITHMIC MODELS
Y 1X
2
log Y log 1 X
2
log 1 log X
2
log 1 2 log X
Y ' 1 2 X '
'
where Y ' log Y ,
X ' log X
1 log 1
'
You thus obtain a linear relationship between Y' and X', as defined. All serious regression
applications allow you to generate logarithmic variables from existing ones.
23
Slide 25
ELASTICITIES AND LOGARITHMIC MODELS
Y 1X
2
log Y log 1 X
2
log 1 log X
2
log 1 2 log X
Y ' 1 2 X '
'
where Y ' log Y ,
X ' log X
1 log 1
'
The coefficient of X' will be a direct estimate of the elasticity, 2.
24
Slide 26
ELASTICITIES AND LOGARITHMIC MODELS
Y 1X
2
log Y log 1 X
2
log 1 log X
2
log 1 2 log X
Y ' 1 2 X '
'
where Y ' log Y ,
X ' log X
1 log 1
'
The constant term will be an estimate of log 1. To obtain an estimate of 1, you calculate
exp(b1'), where b1' is the estimate of 1'. (This assumes that you have used natural
logarithms, that is, logarithms to base e, to transform the model.)
25
Slide 27
ELASTICITIES AND LOGARITHMIC MODELS
16000
FDHO
14000
12000
10000
8000
6000
4000
2000
0
0
20000
40000
60000
80000
EXP
100000 120000 140000 160000
Here is a scatter diagram showing annual household expenditure on FDHO, food eaten at
home, and EXP, total annual household expenditure, both measured in dollars, for 1995 for
a sample of 869 households in the United States (Consumer Expenditure Survey data).
26
Slide 28
ELASTICITIES AND LOGARITHMIC MODELS
. reg FDHO EXP
Source |
SS
df
MS
---------+-----------------------------Model |
915843574
1
915843574
Residual | 2.0815e+09
867 2400831.16
---------+-----------------------------Total | 2.9974e+09
868 3453184.55
Number of obs
F( 1,
867)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
869
381.47
0.0000
0.3055
0.3047
1549.5
-----------------------------------------------------------------------------FDHO |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
---------+-------------------------------------------------------------------EXP |
.0528427
.0027055
19.531
0.000
.0475325
.0581529
_cons |
1916.143
96.54591
19.847
0.000
1726.652
2105.634
------------------------------------------------------------------------------
Here is a regression of FDHO on EXP. It is usual to relate types of consumer expenditure to
total expenditure, rather than income, when using household data. Household income data
tend to be relatively erratic.
27
Slide 29
ELASTICITIES AND LOGARITHMIC MODELS
. reg FDHO EXP
Source |
SS
df
MS
---------+-----------------------------Model |
915843574
1
915843574
Residual | 2.0815e+09
867 2400831.16
---------+-----------------------------Total | 2.9974e+09
868 3453184.55
Number of obs
F( 1,
867)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
869
381.47
0.0000
0.3055
0.3047
1549.5
-----------------------------------------------------------------------------FDHO |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
---------+-------------------------------------------------------------------EXP |
.0528427
.0027055
19.531
0.000
.0475325
.0581529
_cons |
1916.143
96.54591
19.847
0.000
1726.652
2105.634
------------------------------------------------------------------------------
The regression implies that, at the margin, 5 cents out of each dollar of expenditure is spent
on food at home. Does this seem plausible? Probably, though possibly a little low.
28
Slide 30
ELASTICITIES AND LOGARITHMIC MODELS
. reg FDHO EXP
Source |
SS
df
MS
---------+-----------------------------Model |
915843574
1
915843574
Residual | 2.0815e+09
867 2400831.16
---------+-----------------------------Total | 2.9974e+09
868 3453184.55
Number of obs
F( 1,
867)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
869
381.47
0.0000
0.3055
0.3047
1549.5
-----------------------------------------------------------------------------FDHO |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
---------+-------------------------------------------------------------------EXP |
.0528427
.0027055
19.531
0.000
.0475325
.0581529
_cons |
1916.143
96.54591
19.847
0.000
1726.652
2105.634
------------------------------------------------------------------------------
It also suggests that $1,916 would be spent on food at home if total expenditure were zero.
Obviously this is impossible. It might be possible to interpret it somehow as baseline
expenditure, but we would need to take into account family size and composition.
29
Slide 31
ELASTICITIES AND LOGARITHMIC MODELS
16000
FDHO
14000
12000
10000
8000
6000
4000
2000
0
0
20000
40000
60000
80000
100000 120000 140000 160000
EXP
Here is the regression line plotted on the scatter diagram
30
Slide 32
ELASTICITIES AND LOGARITHMIC MODELS
10.00
LGFDHO
9.00
8.00
7.00
6.00
5.00
7.00
8.00
9.00
10.00
11.00
12.00
LGEXP
13.00
We will now fit a constant elasticity function using the same data. The scatter diagram
shows the logarithm of FDHO plotted against the logarithm of EXP.
31
Slide 33
ELASTICITIES AND LOGARITHMIC MODELS
. g LGFDHO = ln(FDHO)
. g LGEXP = ln(EXP)
. reg LGFDHO LGEXP
Source |
SS
df
MS
---------+-----------------------------Model | 84.4161692
1 84.4161692
Residual | 184.579612
866 .213140429
---------+-----------------------------Total | 268.995781
867 .310260416
Number of obs
F( 1,
866)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
868
396.06
0.0000
0.3138
0.3130
.46167
-----------------------------------------------------------------------------LGFDHO |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
---------+-------------------------------------------------------------------LGEXP |
.4800417
.0241212
19.901
0.000
.4326988
.5273846
_cons |
3.166271
.244297
12.961
0.000
2.686787
3.645754
------------------------------------------------------------------------------
Here is the result of regressing LGFDHO on LGEXP. The first two commands generate the
logarithmic variables.
32
Slide 34
ELASTICITIES AND LOGARITHMIC MODELS
. g LGFDHO = ln(FDHO)
. g LGEXP = ln(EXP)
. reg LGFDHO LGEXP
Source |
SS
df
MS
---------+-----------------------------Model | 84.4161692
1 84.4161692
Residual | 184.579612
866 .213140429
---------+-----------------------------Total | 268.995781
867 .310260416
Number of obs
F( 1,
866)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
868
396.06
0.0000
0.3138
0.3130
.46167
-----------------------------------------------------------------------------LGFDHO |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
---------+-------------------------------------------------------------------LGEXP |
.4800417
.0241212
19.901
0.000
.4326988
.5273846
_cons |
3.166271
.244297
12.961
0.000
2.686787
3.645754
------------------------------------------------------------------------------
The estimate of the elasticity is 0.48. Does this seem plausible?
33
Slide 35
ELASTICITIES AND LOGARITHMIC MODELS
. g LGFDHO = ln(FDHO)
. g LGEXP = ln(EXP)
. reg LGFDHO LGEXP
Source |
SS
df
MS
---------+-----------------------------Model | 84.4161692
1 84.4161692
Residual | 184.579612
866 .213140429
---------+-----------------------------Total | 268.995781
867 .310260416
Number of obs
F( 1,
866)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
868
396.06
0.0000
0.3138
0.3130
.46167
-----------------------------------------------------------------------------LGFDHO |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
---------+-------------------------------------------------------------------LGEXP |
.4800417
.0241212
19.901
0.000
.4326988
.5273846
_cons |
3.166271
.244297
12.961
0.000
2.686787
3.645754
------------------------------------------------------------------------------
Yes, definitely. Food is a normal good, so its elasticity should be positive, but it is a basic
necessity. Expenditure on it should grow less rapidly than expenditure generally, so its
elasticity should be less than 1.
34
Slide 36
ELASTICITIES AND LOGARITHMIC MODELS
. g LGFDHO = ln(FDHO)
. g LGEXP = ln(EXP)
. reg LGFDHO LGEXP
Source |
SS
df
MS
---------+-----------------------------Model | 84.4161692
1 84.4161692
Residual | 184.579612
866 .213140429
---------+-----------------------------Total | 268.995781
867 .310260416
Number of obs
F( 1,
866)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
868
396.06
0.0000
0.3138
0.3130
.46167
-----------------------------------------------------------------------------LGFDHO |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
---------+-------------------------------------------------------------------LGEXP |
.4800417
.0241212
19.901
0.000
.4326988
.5273846
_cons |
3.166271
.244297
12.961
0.000
2.686787
3.645754
------------------------------------------------------------------------------
LGF Dˆ HO 3 . 17 0 . 48 LGEXP
FD Hˆ O 23 . 8 EXP
0 . 48
The intercept has no substantive meaning. To obtain an estimate of 1, we calculate e3.16,
which is 23.8.
35
Slide 37
ELASTICITIES AND LOGARITHMIC MODELS
10.00
LGFDHO
9.00
8.00
7.00
6.00
5.00
7.00
8.00
9.00
10.00
11.00
12.00
13.00
LGEXP
Here is the scatter diagram with the regression line plotted.
36
Slide 38
ELASTICITIES AND LOGARITHMIC MODELS
16000
FDHO
14000
12000
10000
8000
6000
4000
2000
0
0
20000
40000
60000
80000
100000 120000 140000 160000
EXP
Here is the regression line from the logarithmic regression plotted in the original scatter
diagram, together with the linear regression line for comparison.
37
Slide 39
ELASTICITIES AND LOGARITHMIC MODELS
16000
FDHO
14000
12000
10000
8000
6000
4000
2000
0
0
20000
40000
60000
80000
100000 120000 140000 160000
EXP
You can see that the logarithmic regression line gives a somewhat better fit, especially at
low levels of expenditure.
38
Slide 40
ELASTICITIES AND LOGARITHMIC MODELS
16000
FDHO
14000
12000
10000
8000
6000
4000
2000
0
0
20000
40000
60000
80000
100000 120000 140000 160000
EXP
However, the difference in the fit is not dramatic. The main reason for preferring the
constant elasticity model is that it makes more sense theoretically. It also has a technical
advantage that we will come to later on (when discussing heteroscedasticity).
39
Slide 41
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: elasticities and logarithmic models
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/
Slide 2
ELASTICITIES AND LOGARITHMIC MODELS
Elasticity of Y with respect to
X
is the proportional change in Y
per proportional change in X:
elasticity
dY Y
dX X
A
Y
dY dX
Y X
Y
X
O0
X
52
This sequence defines elasticities and shows how one may fit nonlinear models with
constant elasticities. First, the general definition of an elasticity.
1
Slide 3
ELASTICITIES AND LOGARITHMIC MODELS
Elasticity of Y with respect to
X
is the proportional change in Y
per proportional change in X:
elasticity
dY Y
dX X
A
Y
dY dX
Y X
slope of the tangent at A
slope of OA
Y
X
O0
X
52
Re-arranging the expression for the elasticity, we can obtain a graphical interpretation.
2
Slide 4
ELASTICITIES AND LOGARITHMIC MODELS
Elasticity of Y with respect to
X
is the proportional change in Y
per proportional change in X:
elasticity
dY Y
dX X
A
Y
dY dX
Y X
slope of the tangent at A
slope of OA
Y
X
O0
X
52
The elasticity at any point on the curve is the ratio of the slope of the tangent at that point to
the slope of the line joining the point to the origin.
3
Slide 5
ELASTICITIES AND LOGARITHMIC MODELS
Elasticity of Y with respect to
X
is the proportional change in Y
per proportional change in X:
elasticity
dY Y
dX X
A
Y
dY dX
Y X
slope of the tangent at A
slope of OA
Y
X
O0
X
52
elasticity < 1
In this case it is clear that the tangent at A is flatter than the line OA and so the elasticity
must be less than 1.
4
Slide 6
ELASTICITIES AND LOGARITHMIC MODELS
Elasticity of Y with respect to
X
is the proportional change in Y
per proportional change in X:
elasticity
dY Y
dX X
A
Y
dY dX
Y X
slope of the tangent at A
slope of OA
O0
X
52
elasticity > 1
In this case the tangent at A is steeper than OA and the elasticity is greater than 1.
5
Slide 7
ELASTICITIES AND LOGARITHMIC MODELS
Y 1 2X
elasticity
Y
A
dY dX
Y X
slope of the tangent at A
slope of OA
2
(1 2 X ) / X
O
Xx
2
(1 / X ) 2
In general the elasticity will be different at different points on the function relating Y to X.
6
Slide 8
ELASTICITIES AND LOGARITHMIC MODELS
Y 1 2X
elasticity
Y
A
dY dX
Y X
slope of the tangent at A
slope of OA
2
(1 2 X ) / X
O
Xx
2
(1 / X ) 2
In the example above, Y is a linear function of X.
7
Slide 9
ELASTICITIES AND LOGARITHMIC MODELS
Y 1 2X
elasticity
Y
A
dY dX
Y X
slope of the tangent at A
slope of OA
2
(1 2 X ) / X
O
Xx
2
(1 / X ) 2
The tangent at any point is coincidental with the line itself, so in this case its slope is always
2. The elasticity depends on the slope of the line joining the point to the origin.
8
Slide 10
ELASTICITIES AND LOGARITHMIC MODELS
Y 1 2X
Y
B
elasticity
A
dY dX
Y X
slope of the tangent at A
slope of OA
2
(1 2 X ) / X
O
Xx
2
(1 / X ) 2
OB is flatter than OA, so the elasticity is greater at B than at A. (This ties in with the
mathematical expression: (1 / X) + 2 is smaller at B than at A, assuming that 1 is positive.)
9
Slide 11
ELASTICITIES AND LOGARITHMIC MODELS
Y 1X
2
However, a function of the type shown above has the same elasticity for all values of X.
10
Slide 12
ELASTICITIES AND LOGARITHMIC MODELS
Y 1X
dY
dX
2
1 2 X
2 1
For the numerator of the elasticity expression, we need the derivative of Y with respect to X.
11
Slide 13
ELASTICITIES AND LOGARITHMIC MODELS
Y 1X
dY
dX
Y
X
2
1 2 X
1X
X
2
2 1
1X
2 1
For the denominator, we need Y/X.
12
Slide 14
ELASTICITIES AND LOGARITHMIC MODELS
Y 1X
dY
dX
Y
X
elasticity
2
1 2 X
1X
2
X
dY dX
Y X
2 1
1X
2 1
1 2 X
1X
2 1
2 1
2
Hence we obtain the expression for the elasticity. This simplifies to 2 and is therefore
constant.
13
Slide 15
ELASTICITIES AND LOGARITHMIC MODELS
Y
Y 1X
2
2 0 . 25
X
By way of illustration, the function will be plotted for a range of values of 2. We will start
with a very low value, 0.25.
14
Slide 16
ELASTICITIES AND LOGARITHMIC MODELS
Y
Y 1X
2
2 0 . 50
X
We will increase 2 in steps of 0.25 and see how the shape of the function changes.
15
Slide 17
ELASTICITIES AND LOGARITHMIC MODELS
Y
Y 1X
2
2 0 . 75
X
16
Slide 18
ELASTICITIES AND LOGARITHMIC MODELS
Y
Y 1X
2
2 1 . 00
X
When 2 is equal to 1, the curve becomes a straight line through the origin.
17
Slide 19
ELASTICITIES AND LOGARITHMIC MODELS
Y
Y 1X
2
2 1 . 25
X
18
Slide 20
ELASTICITIES AND LOGARITHMIC MODELS
Y
Y 1X
2
2 1 . 50
X
19
Slide 21
ELASTICITIES AND LOGARITHMIC MODELS
Y
Y 1X
2
2 1 . 75
X
Note that the curvature can be quite gentle over wide ranges of X.
20
Slide 22
ELASTICITIES AND LOGARITHMIC MODELS
Y
Y 1X
2
2 2 . 00
X
This means that even if the true model is of the constant elasticity form, a linear model may
be a good approximation over a limited range.
21
Slide 23
ELASTICITIES AND LOGARITHMIC MODELS
Y 1X
2
log Y log 1 X
2
log 1 log X
2
log 1 2 log X
It is easy to fit a constant elasticity function using a sample of observations. You can
linearize the model by taking the logarithms of both sides.
22
Slide 24
ELASTICITIES AND LOGARITHMIC MODELS
Y 1X
2
log Y log 1 X
2
log 1 log X
2
log 1 2 log X
Y ' 1 2 X '
'
where Y ' log Y ,
X ' log X
1 log 1
'
You thus obtain a linear relationship between Y' and X', as defined. All serious regression
applications allow you to generate logarithmic variables from existing ones.
23
Slide 25
ELASTICITIES AND LOGARITHMIC MODELS
Y 1X
2
log Y log 1 X
2
log 1 log X
2
log 1 2 log X
Y ' 1 2 X '
'
where Y ' log Y ,
X ' log X
1 log 1
'
The coefficient of X' will be a direct estimate of the elasticity, 2.
24
Slide 26
ELASTICITIES AND LOGARITHMIC MODELS
Y 1X
2
log Y log 1 X
2
log 1 log X
2
log 1 2 log X
Y ' 1 2 X '
'
where Y ' log Y ,
X ' log X
1 log 1
'
The constant term will be an estimate of log 1. To obtain an estimate of 1, you calculate
exp(b1'), where b1' is the estimate of 1'. (This assumes that you have used natural
logarithms, that is, logarithms to base e, to transform the model.)
25
Slide 27
ELASTICITIES AND LOGARITHMIC MODELS
16000
FDHO
14000
12000
10000
8000
6000
4000
2000
0
0
20000
40000
60000
80000
EXP
100000 120000 140000 160000
Here is a scatter diagram showing annual household expenditure on FDHO, food eaten at
home, and EXP, total annual household expenditure, both measured in dollars, for 1995 for
a sample of 869 households in the United States (Consumer Expenditure Survey data).
26
Slide 28
ELASTICITIES AND LOGARITHMIC MODELS
. reg FDHO EXP
Source |
SS
df
MS
---------+-----------------------------Model |
915843574
1
915843574
Residual | 2.0815e+09
867 2400831.16
---------+-----------------------------Total | 2.9974e+09
868 3453184.55
Number of obs
F( 1,
867)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
869
381.47
0.0000
0.3055
0.3047
1549.5
-----------------------------------------------------------------------------FDHO |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
---------+-------------------------------------------------------------------EXP |
.0528427
.0027055
19.531
0.000
.0475325
.0581529
_cons |
1916.143
96.54591
19.847
0.000
1726.652
2105.634
------------------------------------------------------------------------------
Here is a regression of FDHO on EXP. It is usual to relate types of consumer expenditure to
total expenditure, rather than income, when using household data. Household income data
tend to be relatively erratic.
27
Slide 29
ELASTICITIES AND LOGARITHMIC MODELS
. reg FDHO EXP
Source |
SS
df
MS
---------+-----------------------------Model |
915843574
1
915843574
Residual | 2.0815e+09
867 2400831.16
---------+-----------------------------Total | 2.9974e+09
868 3453184.55
Number of obs
F( 1,
867)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
869
381.47
0.0000
0.3055
0.3047
1549.5
-----------------------------------------------------------------------------FDHO |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
---------+-------------------------------------------------------------------EXP |
.0528427
.0027055
19.531
0.000
.0475325
.0581529
_cons |
1916.143
96.54591
19.847
0.000
1726.652
2105.634
------------------------------------------------------------------------------
The regression implies that, at the margin, 5 cents out of each dollar of expenditure is spent
on food at home. Does this seem plausible? Probably, though possibly a little low.
28
Slide 30
ELASTICITIES AND LOGARITHMIC MODELS
. reg FDHO EXP
Source |
SS
df
MS
---------+-----------------------------Model |
915843574
1
915843574
Residual | 2.0815e+09
867 2400831.16
---------+-----------------------------Total | 2.9974e+09
868 3453184.55
Number of obs
F( 1,
867)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
869
381.47
0.0000
0.3055
0.3047
1549.5
-----------------------------------------------------------------------------FDHO |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
---------+-------------------------------------------------------------------EXP |
.0528427
.0027055
19.531
0.000
.0475325
.0581529
_cons |
1916.143
96.54591
19.847
0.000
1726.652
2105.634
------------------------------------------------------------------------------
It also suggests that $1,916 would be spent on food at home if total expenditure were zero.
Obviously this is impossible. It might be possible to interpret it somehow as baseline
expenditure, but we would need to take into account family size and composition.
29
Slide 31
ELASTICITIES AND LOGARITHMIC MODELS
16000
FDHO
14000
12000
10000
8000
6000
4000
2000
0
0
20000
40000
60000
80000
100000 120000 140000 160000
EXP
Here is the regression line plotted on the scatter diagram
30
Slide 32
ELASTICITIES AND LOGARITHMIC MODELS
10.00
LGFDHO
9.00
8.00
7.00
6.00
5.00
7.00
8.00
9.00
10.00
11.00
12.00
LGEXP
13.00
We will now fit a constant elasticity function using the same data. The scatter diagram
shows the logarithm of FDHO plotted against the logarithm of EXP.
31
Slide 33
ELASTICITIES AND LOGARITHMIC MODELS
. g LGFDHO = ln(FDHO)
. g LGEXP = ln(EXP)
. reg LGFDHO LGEXP
Source |
SS
df
MS
---------+-----------------------------Model | 84.4161692
1 84.4161692
Residual | 184.579612
866 .213140429
---------+-----------------------------Total | 268.995781
867 .310260416
Number of obs
F( 1,
866)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
868
396.06
0.0000
0.3138
0.3130
.46167
-----------------------------------------------------------------------------LGFDHO |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
---------+-------------------------------------------------------------------LGEXP |
.4800417
.0241212
19.901
0.000
.4326988
.5273846
_cons |
3.166271
.244297
12.961
0.000
2.686787
3.645754
------------------------------------------------------------------------------
Here is the result of regressing LGFDHO on LGEXP. The first two commands generate the
logarithmic variables.
32
Slide 34
ELASTICITIES AND LOGARITHMIC MODELS
. g LGFDHO = ln(FDHO)
. g LGEXP = ln(EXP)
. reg LGFDHO LGEXP
Source |
SS
df
MS
---------+-----------------------------Model | 84.4161692
1 84.4161692
Residual | 184.579612
866 .213140429
---------+-----------------------------Total | 268.995781
867 .310260416
Number of obs
F( 1,
866)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
868
396.06
0.0000
0.3138
0.3130
.46167
-----------------------------------------------------------------------------LGFDHO |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
---------+-------------------------------------------------------------------LGEXP |
.4800417
.0241212
19.901
0.000
.4326988
.5273846
_cons |
3.166271
.244297
12.961
0.000
2.686787
3.645754
------------------------------------------------------------------------------
The estimate of the elasticity is 0.48. Does this seem plausible?
33
Slide 35
ELASTICITIES AND LOGARITHMIC MODELS
. g LGFDHO = ln(FDHO)
. g LGEXP = ln(EXP)
. reg LGFDHO LGEXP
Source |
SS
df
MS
---------+-----------------------------Model | 84.4161692
1 84.4161692
Residual | 184.579612
866 .213140429
---------+-----------------------------Total | 268.995781
867 .310260416
Number of obs
F( 1,
866)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
868
396.06
0.0000
0.3138
0.3130
.46167
-----------------------------------------------------------------------------LGFDHO |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
---------+-------------------------------------------------------------------LGEXP |
.4800417
.0241212
19.901
0.000
.4326988
.5273846
_cons |
3.166271
.244297
12.961
0.000
2.686787
3.645754
------------------------------------------------------------------------------
Yes, definitely. Food is a normal good, so its elasticity should be positive, but it is a basic
necessity. Expenditure on it should grow less rapidly than expenditure generally, so its
elasticity should be less than 1.
34
Slide 36
ELASTICITIES AND LOGARITHMIC MODELS
. g LGFDHO = ln(FDHO)
. g LGEXP = ln(EXP)
. reg LGFDHO LGEXP
Source |
SS
df
MS
---------+-----------------------------Model | 84.4161692
1 84.4161692
Residual | 184.579612
866 .213140429
---------+-----------------------------Total | 268.995781
867 .310260416
Number of obs
F( 1,
866)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
868
396.06
0.0000
0.3138
0.3130
.46167
-----------------------------------------------------------------------------LGFDHO |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
---------+-------------------------------------------------------------------LGEXP |
.4800417
.0241212
19.901
0.000
.4326988
.5273846
_cons |
3.166271
.244297
12.961
0.000
2.686787
3.645754
------------------------------------------------------------------------------
LGF Dˆ HO 3 . 17 0 . 48 LGEXP
FD Hˆ O 23 . 8 EXP
0 . 48
The intercept has no substantive meaning. To obtain an estimate of 1, we calculate e3.16,
which is 23.8.
35
Slide 37
ELASTICITIES AND LOGARITHMIC MODELS
10.00
LGFDHO
9.00
8.00
7.00
6.00
5.00
7.00
8.00
9.00
10.00
11.00
12.00
13.00
LGEXP
Here is the scatter diagram with the regression line plotted.
36
Slide 38
ELASTICITIES AND LOGARITHMIC MODELS
16000
FDHO
14000
12000
10000
8000
6000
4000
2000
0
0
20000
40000
60000
80000
100000 120000 140000 160000
EXP
Here is the regression line from the logarithmic regression plotted in the original scatter
diagram, together with the linear regression line for comparison.
37
Slide 39
ELASTICITIES AND LOGARITHMIC MODELS
16000
FDHO
14000
12000
10000
8000
6000
4000
2000
0
0
20000
40000
60000
80000
100000 120000 140000 160000
EXP
You can see that the logarithmic regression line gives a somewhat better fit, especially at
low levels of expenditure.
38
Slide 40
ELASTICITIES AND LOGARITHMIC MODELS
16000
FDHO
14000
12000
10000
8000
6000
4000
2000
0
0
20000
40000
60000
80000
100000 120000 140000 160000
EXP
However, the difference in the fit is not dramatic. The main reason for preferring the
constant elasticity model is that it makes more sense theoretically. It also has a technical
advantage that we will come to later on (when discussing heteroscedasticity).
39
Slide 41
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