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Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 7)
Slideshow: White test for heteroscedasticity
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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WHITE TEST FOR HETEROSCEDASTICITY
. reg MANU GDP
Source |
SS
df
MS
-------------+-----------------------------Model | 1.1600e+11
1 1.1600e+11
Residual | 1.4312e+10
26
550462775
-------------+-----------------------------Total | 1.3031e+11
27 4.8264e+09
Number of obs
F( 1,
26)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
28
210.73
0.0000
0.8902
0.8859
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
------------------------------------------------------------------------------
The White test for heteroscedasticity looks for evidence of an association between the
variance of the disturbance term and the regressors without assuming any specific
relationship.
1
WHITE TEST FOR HETEROSCEDASTICITY
. reg MANU GDP
Source |
SS
df
MS
-------------+-----------------------------Model | 1.1600e+11
1 1.1600e+11
Residual | 1.4312e+10
26
550462775
-------------+-----------------------------Total | 1.3031e+11
27 4.8264e+09
Number of obs
F( 1,
26)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
28
210.73
0.0000
0.8902
0.8859
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
------------------------------------------------------------------------------
Since the variance of the disturbance term in observation i is unobservable, the squared
residual for that observation is used as a proxy.
2
WHITE TEST FOR HETEROSCEDASTICITY
. reg MANU GDP
Source |
SS
df
MS
-------------+-----------------------------Model | 1.1600e+11
1 1.1600e+11
Residual | 1.4312e+10
26
550462775
-------------+-----------------------------Total | 1.3031e+11
27 4.8264e+09
Number of obs
F( 1,
26)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
28
210.73
0.0000
0.8902
0.8859
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
-----------------------------------------------------------------------------. predict EMANU, resid
. gen EMANUSQ = EMANU*EMANU
We will perform the test using the manufacturing and GDP data used to illustrate the
Goldfeld–Quandt test. We have regressed MANU on GDP and have saved the residuals as
EMANU. We define EMANUSQ to be the squared residual.
3
WHITE TEST FOR HETEROSCEDASTICITY
. gen GDPSQ = GDP*GDP
. reg EMANUSQ GDP GDPSQ
Source |
SS
df
MS
-------------+-----------------------------Model | 1.3183e+19
2 6.5913e+18
Residual | 4.9179e+19
25 1.9671e+18
-------------+-----------------------------Total | 6.2361e+19
27 2.3097e+18
Number of obs
F( 2,
25)
Prob > F
R-squared
Adj R-squared
Root MSE
=
28
=
3.35
= 0.0514
= 0.2114
= 0.1483
= 1.4e+09
-----------------------------------------------------------------------------EMANUSQ |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------GDP |
6271.896
2758.253
2.27
0.032
591.1687
11952.62
GDPSQ | -.0041155
.0022626
-1.82
0.081
-.0087754
.0005444
_cons | -4.21e+08
4.51e+08
-0.93
0.359
-1.35e+09
5.08e+08
------------------------------------------------------------------------------
Test regression: regress squared residuals on the explanatory variables
in the model, their squares, and their cross-products, omitting any
duplicative variables.
The test consists of regressing the squared residuals on the explanatory variables in the
model, their squares, and their cross-products, omitting any duplicative variables. (For
example, the square of a dummy variable would be duplicative.)
4
WHITE TEST FOR HETEROSCEDASTICITY
. gen GDPSQ = GDP*GDP
. reg EMANUSQ GDP GDPSQ
Source |
SS
df
MS
-------------+-----------------------------Model | 1.3183e+19
2 6.5913e+18
Residual | 4.9179e+19
25 1.9671e+18
-------------+-----------------------------Total | 6.2361e+19
27 2.3097e+18
Number of obs
F( 2,
25)
Prob > F
R-squared
Adj R-squared
Root MSE
=
28
=
3.35
= 0.0514
= 0.2114
= 0.1483
= 1.4e+09
-----------------------------------------------------------------------------EMANUSQ |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------GDP |
6271.896
2758.253
2.27
0.032
591.1687
11952.62
GDPSQ | -.0041155
.0022626
-1.82
0.081
-.0087754
.0005444
_cons | -4.21e+08
4.51e+08
-0.93
0.359
-1.35e+09
5.08e+08
------------------------------------------------------------------------------
Test regression: regress squared residuals on the explanatory variables
in the model, their squares, and their cross-products, omitting any
duplicative variables.
In the present case we regress EMANUSQ on GDP and its square (and a constant).
5
WHITE TEST FOR HETEROSCEDASTICITY
. gen GDPSQ = GDP*GDP
. reg EMANUSQ GDP GDPSQ
Source |
SS
df
MS
-------------+-----------------------------Model | 1.3183e+19
2 6.5913e+18
Residual | 4.9179e+19
25 1.9671e+18
-------------+-----------------------------Total | 6.2361e+19
27 2.3097e+18
Number of obs
F( 2,
25)
Prob > F
R-squared
Adj R-squared
Root MSE
=
28
=
3.35
= 0.0514
= 0.2114
= 0.1483
= 1.4e+09
-----------------------------------------------------------------------------EMANUSQ |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------GDP |
6271.896
2758.253
2.27
0.032
591.1687
11952.62
GDPSQ | -.0041155
.0022626
-1.82
0.081
-.0087754
.0005444
_cons | -4.21e+08
4.51e+08
-0.93
0.359
-1.35e+09
5.08e+08
------------------------------------------------------------------------------
Test statistic: nR2, using R2 from this regression. Under H0, chi-squared
statistic with degrees of freedom equal to the number of regressors,
including the constant, minus one, in large samples.
The test statistic is nR2, using R2 from this regression. Under the null hypothesis of no
association, it is distributed as a chi-squared statistic with degrees of freedom equal to the
number of regressors, including the constant, minus one, in large samples.
6
WHITE TEST FOR HETEROSCEDASTICITY
. gen GDPSQ = GDP*GDP
. reg EMANUSQ GDP GDPSQ
Source |
SS
df
MS
-------------+-----------------------------Model | 1.3183e+19
2 6.5913e+18
Residual | 4.9179e+19
25 1.9671e+18
-------------+-----------------------------Total | 6.2361e+19
27 2.3097e+18
Number of obs
F( 2,
25)
Prob > F
R-squared
Adj R-squared
Root MSE
=
28
=
3.35
= 0.0514
= 0.2114
= 0.1483
= 1.4e+09
-----------------------------------------------------------------------------EMANUSQ |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------GDP |
6271.896
2758.253
2.27
0.032
591.1687
11952.62
GDPSQ | -.0041155
.0022626
-1.82
0.081
-.0087754
.0005444
_cons | -4.21e+08
4.51e+08
-0.93
0.359
-1.35e+09
5.08e+08
------------------------------------------------------------------------------
nR2 28 0.2114 5.92
52% ( 2) 5.99
R2 is 0.2114 and n is 28. The test statistic is therefore 5.92. The critical value of chi-squared
with two degrees of freedom is 5.99 at the 5 percent level and so the null hypothesis of
homoscedasticity is not rejected.
7
WHITE TEST FOR HETEROSCEDASTICITY
. gen GDPSQ = GDP*GDP
. reg EMANUSQ GDP GDPSQ
Source |
SS
df
MS
-------------+-----------------------------Model | 1.3183e+19
2 6.5913e+18
Residual | 4.9179e+19
25 1.9671e+18
-------------+-----------------------------Total | 6.2361e+19
27 2.3097e+18
Number of obs
F( 2,
25)
Prob > F
R-squared
Adj R-squared
Root MSE
=
28
=
3.35
= 0.0514
= 0.2114
= 0.1483
= 1.4e+09
-----------------------------------------------------------------------------EMANUSQ |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------GDP |
6271.896
2758.253
2.27
0.032
591.1687
11952.62
GDPSQ | -.0041155
.0022626
-1.82
0.081
-.0087754
.0005444
_cons | -4.21e+08
4.51e+08
-0.93
0.359
-1.35e+09
5.08e+08
------------------------------------------------------------------------------
nR2 28 0.2114 5.92
52% ( 2) 5.99
Why has the White test failed to detect heteroscedasticity when the Goldfeld–Quandt test
concluded that it was present at a high level of significance? One reason is that it is a
large-sample test, and the sample is actually quite small.
8
WHITE TEST FOR HETEROSCEDASTICITY
. gen GDPSQ = GDP*GDP
. reg EMANUSQ GDP GDPSQ
Source |
SS
df
MS
-------------+-----------------------------Model | 1.3183e+19
2 6.5913e+18
Residual | 4.9179e+19
25 1.9671e+18
-------------+-----------------------------Total | 6.2361e+19
27 2.3097e+18
Number of obs
F( 2,
25)
Prob > F
R-squared
Adj R-squared
Root MSE
=
28
=
3.35
= 0.0514
= 0.2114
= 0.1483
= 1.4e+09
-----------------------------------------------------------------------------EMANUSQ |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------GDP |
6271.896
2758.253
2.27
0.032
591.1687
11952.62
GDPSQ | -.0041155
.0022626
-1.82
0.081
-.0087754
.0005444
_cons | -4.21e+08
4.51e+08
-0.93
0.359
-1.35e+09
5.08e+08
------------------------------------------------------------------------------
nR2 28 0.2114 5.92
52% ( 2) 5.99
A second is that the White test tends to have low power — a price that one has to pay for its
generality. These problems can be exacerbated by a loss of degrees of freedom if there are
many explanatory variables in the original model.
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 7.2 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
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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