As an example, we quote the following results due to Greene34:
Y = 832.91 -
1834.2 (Income) + 1587.04 (Income)2
OLS se = (327.3)
t = (2.54)
White se = (460.9)
t = (1.81)
where Y = per capita expenditure on public schools by state in 1979 and Income = per capita income by state in 1979. The sample consisted of 50 states plus Washington, D.C.
As the preceding results show, (White's) heteroscedasticity-corrected standard errors are considerably larger than the OLS standard errors and therefore the estimated t values are much smaller than those obtained by OLS. On the basis of the latter, both the regressors are statistically significant at the 5 percent level, whereas on the basis of White estimators they are not. However, it should be pointed out that White's heteroscedasticity-corrected
33More technically, they are known as heteroscedasticity-consistent covariance matrix estimators.
34William H. Greene, Econometric Analysis, 2d ed., Macmillan, New York, 1993, p. 385.
418 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL
standard errors can be larger or smaller than the uncorrected standard errors.
Since White's heteroscedasticity-consistent estimators of the variances are now available in established regression packages, it is recommended that the reader report them. As Wallace and Silver note:
Generally speaking, it is probably a good idea to use the WHITE option [available in regression programs] routinely, perhaps comparing the output with regular OLS output as a check to see whether heteroscedasticity is a serious problem in a particular set of data.35
Plausible Assumptions about Heteroscedasticity Pattern. Apart from being a large-sample procedure, one drawback of the White procedure is that the estimators thus obtained may not be so efficient as those obtained by methods that transform data to reflect specific types of heteroscedasticity. To illustrate this, let us revert to the two-variable regression model:
We now consider several assumptions about the pattern of heteroscedasticity.
Assumption 1: The error variance is proportional to Xf:
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