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Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 7)
Slideshow: heteroscedasticity-consistent standard errors
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 7). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/133/
Available in LSE Learning Resources Online: May 2012
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HETEROSCEDASTICITY-CONSISTENT STANDARD ERRORS
n
OLS
2
b
where
ai
2 a i ui
i 1
(Xi X )
n
2
(
X
X
)
i
i 1
Heteroscedasticity causes OLS standard errors to be biased is finite samples. However it
can be demonstrated that they are nevertheless consistent, provided that their variances
are distributed independently of the regressors.
1
HETEROSCEDASTICITY-CONSISTENT STANDARD ERRORS
n
OLS
2
b
ai
where
2 a i ui
i 1
(Xi X )
n
2
(
X
X
)
i
i 1
b2
OLS
2
a i2 E ui2 a i2 u2i
n
n
i 1
i 1
Even if this is not the case, it is still possible to obtain consistent estimators. We have seen
that the slope coefficient in a simple OLS regression could be decomposed as above.
2
HETEROSCEDASTICITY-CONSISTENT STANDARD ERRORS
n
OLS
2
b
ai
where
2 a i ui
i 1
(Xi X )
n
2
(
X
X
)
i
i 1
b2
OLS
2
a i2 E ui2 a i2 u2i
n
n
i 1
i 1
We have also seen that the variance of the estimator is given by the expression above if ui
is distributed independently of uj for j i.
3
HETEROSCEDASTICITY-CONSISTENT STANDARD ERRORS
n
2 a i ui
OLS
2
b
i 1
ai
where
(Xi X )
n
2
(
X
X
)
i
i 1
b2
OLS
2
a i2 E ui2 a i2 u2i
n
n
i 1
i 1
n
sb2OLS a i2 ei2
2
i 1
White (1980) demonstrates that a consistent estimator of b2OLS is obtained if the squared
2
residual in observation i is used as an estimator of u2i . Taking the square root, one obtains
a heteroscedasticity-consistent standard error.
4
b2OLS
2
HETEROSCEDASTICITY-CONSISTENT STANDARD ERRORS
n
2 a i ui
OLS
2
b
i 1
ai
where
(Xi X )
n
2
(
X
X
)
i
i 1
b2
OLS
2
a i2 E ui2 a i2 u2i
n
n
i 1
i 1
n
sb2OLS a i2 ei2
2
i 1
Thus in a situation where heteroscedasticity is suspected, but there is not enough
information to identify its nature, it is possible to overcome the problem of biased standard
errors, at least in large samples, and the t tests and F tests are asymptotically valid.
5
b2OLS
2
HETEROSCEDASTICITY-CONSISTENT STANDARD ERRORS
n
2 a i ui
OLS
2
b
i 1
ai
where
(Xi X )
n
2
(
X
X
)
i
i 1
b2
OLS
2
a i2 E ui2 a i2 u2i
n
n
i 1
i 1
n
sb2OLS a i2 ei2
2
i 1
Two points, need to be kept in mind, however. One is that, although the White estimator is
consistent, it may not perform well in finite samples (MacKinnon and White, 1985). The
other is that the OLS estimators remain inefficient.
6
b2OLS
2
HETEROSCEDASTICITY-CONSISTENT STANDARD ERRORS
. reg manu gdp
Source |
SS
df
MS
Number of obs =
28
-------------+-----------------------------F( 1,
26) = 210.73
Model | 1.1600e+11
1 1.1600e+11
Prob > F
= 0.0000
Residual | 1.4312e+10
26
550462775
R-squared
= 0.8902
-------------+-----------------------------Adj R-squared = 0.8859
Total | 1.3031e+11
27 4.8264e+09
Root MSE
=
23462
-----------------------------------------------------------------------------manu |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------gdp |
.193693
.0133428
14.52
0.000
.1662665
.2211195
_cons |
603.9453
5699.677
0.11
0.916
-11111.91
12319.8
. reg manu gdp, robust
Regression with robust standard errors
Number of obs =
28
F( 1,
26) = 116.39
Prob > F
= 0.0000
R-squared
= 0.8902
Root MSE
=
23462
-----------------------------------------------------------------------------|
Robust
manu |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------gdp |
.193693
.0179542
10.79
0.000
.1567877
.2305983
_cons |
603.9453
3542.388
0.17
0.866
-6677.538
7885.429
To illustrate the use of heteroscedasticity-consistent standard errors, the regression of
MANU on GDP in the previous sequence is repeated with the ‘robust’ option available in
Stata.
7
b2OLS
2
HETEROSCEDASTICITY-CONSISTENT STANDARD ERRORS
. reg manu gdp
Source |
SS
df
MS
Number of obs =
28
-------------+-----------------------------F( 1,
26) = 210.73
Model | 1.1600e+11
1 1.1600e+11
Prob > F
= 0.0000
Residual | 1.4312e+10
26
550462775
R-squared
= 0.8902
-------------+-----------------------------Adj R-squared = 0.8859
Total | 1.3031e+11
27 4.8264e+09
Root MSE
=
23462
-----------------------------------------------------------------------------manu |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------gdp |
.193693
.0133428
14.52
0.000
.1662665
.2211195
_cons |
603.9453
5699.677
0.11
0.916
-11111.91
12319.8
. reg manu gdp, robust
Regression with robust standard errors
Number of obs =
28
F( 1,
26) = 116.39
Prob > F
= 0.0000
R-squared
= 0.8902
Root MSE
=
23462
-----------------------------------------------------------------------------|
Robust
manu |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------gdp |
.193693
.0179542
10.79
0.000
.1567877
.2305983
_cons |
603.9453
3542.388
0.17
0.866
-6677.538
7885.429
The point estimates of the coefficients are exactly the same. They are not affected by the
procedure, and so their inefficiency is not alleviated.
8
b2OLS
2
HETEROSCEDASTICITY-CONSISTENT STANDARD ERRORS
. reg manu gdp
Source |
SS
df
MS
Number of obs =
28
-------------+-----------------------------F( 1,
26) = 210.73
Model | 1.1600e+11
1 1.1600e+11
Prob > F
= 0.0000
Residual | 1.4312e+10
26
550462775
R-squared
= 0.8902
-------------+-----------------------------Adj R-squared = 0.8859
Total | 1.3031e+11
27 4.8264e+09
Root MSE
=
23462
-----------------------------------------------------------------------------manu |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------gdp |
.193693
.0133428
14.52
0.000
.1662665
.2211195
_cons |
603.9453
5699.677
0.11
0.916
-11111.91
12319.8
. reg manu gdp, robust
Regression with robust standard errors
Number of obs =
28
F( 1,
26) = 116.39
Prob > F
= 0.0000
R-squared
= 0.8902
Root MSE
=
23462
-----------------------------------------------------------------------------|
Robust
manu |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------gdp |
.193693
.0179542
10.79
0.000
.1567877
.2305983
_cons |
603.9453
3542.388
0.17
0.866
-6677.538
7885.429
However the standard error of the coefficient of GDP rises from 0.13 to 0.18, indicating that
it is underestimated in the original OLS regression.
9
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 7.3 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