This is a good place to remind students that homoskedasticity played no role in showing that
OLS is unbiased for the parameters in the regression equation.
In addition, you probably should
mention that there is nothing wrong with the
-squared or adjusted
-squared as goodness-of-fit
The key is that these are estimates of the population
-squared, 1 – [Var(
where the variances are the
variances in the population.
the adjusted version, consistently estimate the population
-squared whether or not Var(
) depends on
Of course, heteroskedasticity causes the usual standard errors,
statistics to be invalid, even in large samples, with or without normality.
By explicitly stating the homoskedasticity assumption as conditional on the explanatory
variables that appear in the conditional mean, it is clear that only heteroskedasticity that depends
on the explanatory variables in the model affects the validity of standard errors and test statistics.
The version of the Breusch-Pagan test in the text, and the White test, are ideally suited for
detecting forms of heteroskedasticity that invalidate inference obtained under homoskedasticity.
If heteroskedasticity depends on an exogenous variable that does not also appear in the mean
equation, this can be exploited in weighted least squares for efficiency, but only rarely is such a
One case where such a variable is available is when an individual-level
equation has been aggregated.
I discuss this case in the text but I rarely have time to teach it.
As I mention in the text, other traditional tests for heteroskedasticity, such as the Park and
Glejser tests, do not directly test what we want, or add too many assumptions under the null.
The Goldfeld-Quandt test only works when there is a natural way to order the data based on one
This is rare in practice, especially for cross-sectional applications.
Some argue that weighted least squares estimation is a relic, and is no longer necessary given the
availability of heteroskedasticity-robust standard errors and test statistics.
While I am
sympathetic to this argument, it presumes that we do not care much about efficiency.
large samples, the OLS estimates may not be precise enough to learn much about the population
With substantial heteroskedasticity we might do better with weighted least squares,
even if the weighting function is misspecified.
As discussed in the text on pages 287-288, one
can, and probably should, compute robust standard errors after weighted least squares.
asymptotic efficiency comparisons, these would be directly comparable to the
heteroskedasiticity-robust standard errors for OLS.