1. Heteroscedasticity
Heteroscedasticity is present if the variance of the error term
is not a constant.
This is most commonly a problem when dealing with cross-
section data.
A common situation is where the variance is related to one of
the RHS variables.

2. An example of a regression model with heteroscedasticity.
Note how the variance increases as x increases.

3. Consequences of Heteroscedasticity
Heteroscedasticity does not, in itself, mean that OLS will be
biased.
However, it does mean that OLS will be inefficient since one
of the Gauss-Markov assumptions is no longer valid.
If the variance of the error term is positively correlated with
one of the RHS variables then the OLS estimates of the standard
errors will be biased downwards.
If heteroscedasticity is a symptom of some other kind of
misspecification (e.g. omitted variables) then it is possible that
OLS will be biased.

4. Dealing with heteroscedasticity
If we know the form of the heteroscedasticity then it is easy to
deal with. Suppose we have:
Then we can redefine the model as:
The revised model has constant variance and hence OLS is BLUE.
The problem is that we often don’t know the form of the
heteroscedasticity prior to estimation.

5. Testing for heteroscedasticity
The Goldfeld-Quandt test provides a basic test for
heteroscedasticity. This is constructed as follows:
Order the data according to the size of the exogenous variable
we believe is related to the variance of the error term.
2. Divide the sample into three sections of size n, N-2n and n
respectively. n should be approximately equal to 3N/8.
3. Estimate separate regressions for the first and last n
observations and generate the residual sum of squares. Perform
an F test for the equality of these sums of squares

6. For the Goldfeld-Quandt test we divide the sample into
three sections and discard the middle section.
We estimate separate regressions for the lower and upper
sections and compare the RSS’s.

7. Example: The following equation relates profits to employment
and capital for a sample of 474 large UK companies.

8. Next we order the data according to the size of the capital stock and
divide it into three sections of size 178, 118 and 178. We then
discard the middle section and estimate separate regressions for the
lower and upper sections.

9. Finally, we calculate the F statistic and compare it with the
5% critical value.
The 5% critical value for the F distribution with 178 and 178
degrees of freedom is 1.28.
Therefore we reject the null hypothesis that the RSS’s are equal
in favour of the alternative that the RSS is larger for large
values of the capital stock i.e. heteroscedasticity is present in
this model.

10. Problems with the Goldfeld-Quandt test
We need to know which variable to use to order the data
before we perform the test.
2. If the number of observations is small it may be impractical
to divide the sample into three sections and discard the
middle section.

11. White’s test for heteroscedasticity
White’s test uses the residuals from OLS estimation to construct
a test statistic. The procedure is as follows:
Estimate the model by OLS and save the residuals.
2. Regress the squared residuals on the original regressors as
well as their squared values and (possibly) their cross-products.
3. Perform either an F-test or a Chi-squared test for the
significance of the regressors in the stage 2 regression.

12. Example: Using the residuals from our profits equation we
obtain the following auxiliary regression.
Both test statistics indicate significant heteroscedasticity.

13. Including cross-products makes little difference in this case.
In practice it is better to include cross-products unless the number
of regressors becomes so large that we have insufficient degrees
of freedom.