Chapter 6 variable misspecification ii inclusion of an irrelevant
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Transcript Chapter 6 variable misspecification ii inclusion of an irrelevant
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 6)
Slideshow: variable misspecification ii: inclusion of an irrelevant variable
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 6). [Teaching Resource]
© 2012 The Author
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VARIABLE MISSPECIFICATION II: INCLUSION OF AN IRRELEVANT VARIABLE
Consequences of variable misspecification
TRUE MODEL
FITTED MODEL
Y 1 2 X 2 u Y 1 2 X 2 3 X 3 u
Yˆ b1 b2 X 2
Yˆ b1 b2 X 2
b3 X 3
Correct specification,
no problems
Coefficients are biased (in
general). Standard
errors are invalid.
Correct specification,
no problems
In this sequence we will investigate the consequences of including an irrelevant variable in
a regression model.
1
VARIABLE MISSPECIFICATION II: INCLUSION OF AN IRRELEVANT VARIABLE
Consequences of variable misspecification
TRUE MODEL
FITTED MODEL
Y 1 2 X 2 u Y 1 2 X 2 3 X 3 u
Yˆ b1 b2 X 2
Yˆ b1 b2 X 2
b3 X 3
Correct specification,
no problems
Coefficients are
unbiased (in general),
but inefficient.
Standard errors are
valid (in general)
Coefficients are biased (in
general). Standard
errors are invalid.
Correct specification,
no problems
The effects are different from those of omitted variable misspecification. In this case the
coefficients in general remain unbiased, but they are inefficient. The standard errors remain
valid, but are needlessly large.
2
VARIABLE MISSPECIFICATION II: INCLUSION OF AN IRRELEVANT VARIABLE
Y 1 2 X 2 u
Yˆ b1 b2 X 2 b3 X 3
These results can be demonstrated quickly.
3
VARIABLE MISSPECIFICATION II: INCLUSION OF AN IRRELEVANT VARIABLE
Y 1 2 X 2 u
Yˆ b1 b2 X 2 b3 X 3
Y 1 2 X 2 0 X 3 u
Rewrite the true model adding X3 as an explanatory variable, with a coefficient of 0. Now the
true model and the fitted model coincide. Hence b2 will be an unbiased estimator of 2 and
b3 will be an unbiased estimator of 0.
4
VARIABLE MISSPECIFICATION II: INCLUSION OF AN IRRELEVANT VARIABLE
Y 1 2 X 2 u
Yˆ b1 b2 X 2 b3 X 3
Y 1 2 X 2 0 X 3 u
u2
1
2
2
1
r
X
X
X2 ,X3
2i 2
2
b2
However, the variance of b2 will be larger than it would have been if the correct simple
regression had been run because it includes the factor 1 / (1 – r2), where r is the correlation
between X2 and X3.
5
VARIABLE MISSPECIFICATION II: INCLUSION OF AN IRRELEVANT VARIABLE
Y 1 2 X 2 u
Yˆ b1 b2 X 2 b3 X 3
Y 1 2 X 2 0 X 3 u
u2
1
2
2
1
r
X
X
X2 ,X3
2i 2
2
b2
The estimator b2 using the multiple regression model will therefore be less efficient than the
alternative using the simple regression model.
6
VARIABLE MISSPECIFICATION II: INCLUSION OF AN IRRELEVANT VARIABLE
Y 1 2 X 2 u
Yˆ b1 b2 X 2 b3 X 3
Y 1 2 X 2 0 X 3 u
u2
1
2
2
1
r
X
X
X2 ,X3
2i 2
2
b2
The intuitive reason for this is that the simple regression model exploits the information
that X3 should not be in the regression, while with the multiple regression model you find
this out from the regression results.
7
VARIABLE MISSPECIFICATION II: INCLUSION OF AN IRRELEVANT VARIABLE
Y 1 2 X 2 u
Yˆ b1 b2 X 2 b3 X 3
Y 1 2 X 2 0 X 3 u
u2
1
2
2
1
r
X
X
X2 ,X3
2i 2
2
b2
The standard errors remain valid, because the model is formally correctly specified, but
they will tend to be larger than those obtained in a simple regression, reflecting the loss of
efficiency.
8
VARIABLE MISSPECIFICATION II: INCLUSION OF AN IRRELEVANT VARIABLE
Y 1 2 X 2 u
Yˆ b1 b2 X 2 b3 X 3
Y 1 2 X 2 0 X 3 u
u2
1
2
2
1
r
X
X
X2 ,X3
2i 2
2
b2
These are the results in general. Note that if X2 and X3 happen to be uncorrelated, there will
be no loss of efficiency after all.
9
VARIABLE MISSPECIFICATION II: INCLUSION OF AN IRRELEVANT VARIABLE
. reg LGFDHO LGEXP LGSIZE
Source |
SS
df
MS
---------+-----------------------------Model | 138.776549
2 69.3882747
Residual | 130.219231
865 .150542464
---------+-----------------------------Total | 268.995781
867 .310260416
Number of obs
F( 2,
865)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
868
460.92
0.0000
0.5159
0.5148
.388
-----------------------------------------------------------------------------LGFDHO |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
---------+-------------------------------------------------------------------LGEXP |
.2866813
.0226824
12.639
0.000
.2421622
.3312003
LGSIZE |
.4854698
.0255476
19.003
0.000
.4353272
.5356124
_cons |
4.720269
.2209996
21.359
0.000
4.286511
5.154027
------------------------------------------------------------------------------
The analysis will be illustrated using a regression of LGFDHO, the logarithm of annual
household expenditure on food eaten at home, on LGEXP, the logarithm of total annual household
expenditure, and LGSIZE, the logarithm of the number of persons in the household.
10
VARIABLE MISSPECIFICATION II: INCLUSION OF AN IRRELEVANT VARIABLE
. reg LGFDHO LGEXP LGSIZE
Source |
SS
df
MS
---------+-----------------------------Model | 138.776549
2 69.3882747
Residual | 130.219231
865 .150542464
---------+-----------------------------Total | 268.995781
867 .310260416
Number of obs
F( 2,
865)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
868
460.92
0.0000
0.5159
0.5148
.388
-----------------------------------------------------------------------------LGFDHO |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
---------+-------------------------------------------------------------------LGEXP |
.2866813
.0226824
12.639
0.000
.2421622
.3312003
LGSIZE |
.4854698
.0255476
19.003
0.000
.4353272
.5356124
_cons |
4.720269
.2209996
21.359
0.000
4.286511
5.154027
------------------------------------------------------------------------------
The source of the data was the 1995 US Consumer Expenditure Survey. The sample size was 868.
11
VARIABLE MISSPECIFICATION II: INCLUSION OF AN IRRELEVANT VARIABLE
. reg LGFDHO LGEXP LGSIZE LGHOUS
Source |
SS
df
MS
---------+-----------------------------Model | 138.841976
3 46.2806586
Residual | 130.153805
864 .150640978
---------+-----------------------------Total | 268.995781
867 .310260416
Number of obs
F( 3,
864)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
868
307.22
0.0000
0.5161
0.5145
.38812
-----------------------------------------------------------------------------LGFDHO |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
---------+-------------------------------------------------------------------LGEXP |
.2673552
.0370782
7.211
0.000
.1945813
.340129
LGSIZE |
.4868228
.0256383
18.988
0.000
.4365021
.5371434
LGHOUS |
.0229611
.0348408
0.659
0.510
-.0454214
.0913436
_cons |
4.708772
.2217592
21.234
0.000
4.273522
5.144022
------------------------------------------------------------------------------
Now add LGHOUS, the logarithm of annual expenditure on housing services. It is safe to
assume that LGHOUS is an irrelevant variable and, not surprisingly, its coefficient is not
significantly different from zero.
12
VARIABLE MISSPECIFICATION II: INCLUSION OF AN IRRELEVANT VARIABLE
. reg LGFDHO LGEXP LGSIZE LGHOUS
. cor LGHOUS LGEXP LGSIZE
Source |
SS
df
MS
Number of obs =
868
(obs=869)
---------+-----------------------------F( 3,
864) = 307.22
Model | 138.841976
3 46.2806586
Prob > F LGEXP = LGSIZE
0.0000
|
LGHOUS
Residual | 130.153805
864 .150640978 --------+--------------------------R-squared
= 0.5161
---------+-----------------------------Adj R-squared = 0.5145
lGHOUS|
1.0000
Total | 268.995781
867 .310260416
Root MSE1.0000 = .38812
LGEXP|
0.8137
LGSIZE|
0.3256
0.4491
1.0000
-----------------------------------------------------------------------------LGFDHO |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
---------+-------------------------------------------------------------------LGEXP |
.2673552
.0370782
7.211
0.000
.1945813
.340129
LGSIZE |
.4868228
.0256383
18.988
0.000
.4365021
.5371434
LGHOUS |
.0229611
.0348408
0.659
0.510
-.0454214
.0913436
_cons |
4.708772
.2217592
21.234
0.000
4.273522
5.144022
------------------------------------------------------------------------------
It is however highly correlated with LGEXP (correlation coefficient 0.81), and also, to a
lesser extent, with LGSIZE (correlation coefficient 0.33).
13
VARIABLE MISSPECIFICATION II: INCLUSION OF AN IRRELEVANT VARIABLE
. reg LGFDHO LGEXP LGSIZE
-----------------------------------------------------------------------------LGFDHO |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
---------+-------------------------------------------------------------------LGEXP |
.2866813
.0226824
12.639
0.000
.2421622
.3312003
LGSIZE |
.4854698
.0255476
19.003
0.000
.4353272
.5356124
_cons |
4.720269
.2209996
21.359
0.000
4.286511
5.154027
-----------------------------------------------------------------------------. reg LGFDHO LGEXP LGSIZE LGHOUS
-----------------------------------------------------------------------------LGFDHO |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
---------+-------------------------------------------------------------------LGEXP |
.2673552
.0370782
7.211
0.000
.1945813
.340129
LGSIZE |
.4868228
.0256383
18.988
0.000
.4365021
.5371434
LGHOUS |
.0229611
.0348408
0.659
0.510
-.0454214
.0913436
_cons |
4.708772
.2217592
21.234
0.000
4.273522
5.144022
------------------------------------------------------------------------------
Its inclusion does not cause the coefficients of those variables to be biased.
14
VARIABLE MISSPECIFICATION II: INCLUSION OF AN IRRELEVANT VARIABLE
. reg LGFDHO LGEXP LGSIZE
-----------------------------------------------------------------------------LGFDHO |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
---------+-------------------------------------------------------------------LGEXP |
.2866813
.0226824
12.639
0.000
.2421622
.3312003
LGSIZE |
.4854698
.0255476
19.003
0.000
.4353272
.5356124
_cons |
4.720269
.2209996
21.359
0.000
4.286511
5.154027
-----------------------------------------------------------------------------. reg LGFDHO LGEXP LGSIZE LGHOUS
-----------------------------------------------------------------------------LGFDHO |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
---------+-------------------------------------------------------------------LGEXP |
.2673552
.0370782
7.211
0.000
.1945813
.340129
LGSIZE |
.4868228
.0256383
18.988
0.000
.4365021
.5371434
LGHOUS |
.0229611
.0348408
0.659
0.510
-.0454214
.0913436
_cons |
4.708772
.2217592
21.234
0.000
4.273522
5.144022
------------------------------------------------------------------------------
But it does increase their standard errors, particularly that of LGEXP, as you would expect,
reflecting the loss of efficiency.
15
Copyright Christopher Dougherty 2011.
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Introduction to Econometrics, fourth edition 2011, Oxford University Press.
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11.07.25