The McGraw-Hill Series Economics essentials of economics brue, McConnell, and Flynn Essentials of Economics

EXAMPLE 11.6
(
Continued
)
guj75772_ch11.qxd 12/08/2008 07:04 PM Page 388

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Chapter 11
Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
389
you estimate this model, obtain 
ˆ
u
i
from this model, and then estimate
ˆ
u
2
i
=
α
1
+
α
2
(
ˆ
Y
i
)
2
+
v
i
(11.5.27)
where
ˆ
Y
i
are the estimated values from the model (11.5.26). The null hypothesis is that
α
2
=
0
.
If this is not rejected, then one could conclude that there is no heteroscedasticity. The
null hypothesis can be tested by the usual
t
test or the
F
test. (Note that
F
1,
k
=
t
k
2
.
) If the
model (11.5.26) is double log, then the squared residuals are regressed on (log
ˆ
Y
i
)
2
.
One other
advantage of the KB test is that it is applicable even if the error term in the original model
(11.5.26) is not normally distributed. If you apply the KB test to Example 11.1, you will find
that the slope coefficient in the regression of the squared residuals obtained from Eq. (11.5.3)
on the estimated
ˆ
Y
2
i
from Eq. (11.5.3) is statistically not different from zero, thus reinforcing
the Park test. This result should not be surprising since in the present instance we only have a
single regressor. But the KB test is applicable if there is one regressor or many.
A Note Regarding the Tests of Heteroscedasticity
We have discussed several tests of heteroscedasticity in this section. So how do we decide
which is the best test? This is not an easy question to answer, for these tests are based on var-
ious assumptions. In comparing the tests, we need to pay attention to their size (or level of sig-
nificance), power (the probability of rejecting a false hypothesis), and sensitivity to outliers.
We have already pointed out some of the limitations of the popular and easy-to-apply
White’s test of heteroscedasticity. As a result of these limitations, it may have low power
against the alternatives. Besides, the test is of little help in identifying the factors or vari-
ables that cause heteroscedasticity.
Similarly, the Breusch–Pagan–Godfrey test is sensitive to the assumption of normality.
In contrast, the test of Koenker–Bassett does not rely on the normality assumption and
may therefore be more powerful.
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In the Goldfeld–Quandt test if we omit too many
observations, we may diminish the power of the test.
It is beyond the scope of this text to provide a comparative analysis of the various
heteroscedasticity tests. But the interested reader may refer to the article by John Lyon and
Chin-Ling Tsai to get some idea about the strengths and weaknesses of the various tests of
heteroscedasticity.
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