Notes on heteroskedasticity assignments

•          After correcting for heteroskedasticity using GLS/WLS, some attempted to test whether any heteroskedasticity remained by entering the DIAG /HET command. This does not work, because this command implements heteroskedasticity tests on the untransformed (unweighted) residuals. Therefore, the tests should reflect the presence of heteroskedasticity, because while GLS/WLS “corrects” the estimates of the unknown parameters for heteroskedasticity (more precisely, provides an estimate that is asymptotically efficient), it does not remove the heteroskedasticity from the untransformed residuals (although it does so from the weighted residuals). Therefore, a DIAG/HET should not reject the null of homoskedasticity at this stage; if it does so, this would cast doubt on whether the GLS/WLS procedure was appropriate in the first place.

•          On the other hand, the residuals saved by the RESID= option in WLS are the transformed (weighted) residuals, and should be homoskedastic if WLS has fully corrected for heteroskedasticity. One way of testing for this is to estimate an auxiliary equation using these transformed residuals, and using NR² as the test statistic, which should be asymptotically distributed as chi-square.

•          The residual statistics reported in a WLS regression (for example VARIANCE OF THE ESTIMATE-SIGMA**2, STANDARD ERROR OF THE ESTIMATE-SIGMA, and SUM OF SQUARED ERRORS-SSE) are all based on the transformed (weighted) residuals. So therefore, are the Model Selection test statistics reported in the ANOVA option. This is not necessarily the wrong thing to do, but it should be understood that in any model comparisons, the high-variance observations are systematically underweighted by this procedure. This may be a good thing, if you want to avoid having these observations dominate the model selection comparisons. But if you want model selection to be based on how well the alternative models fit the original (untransformed) data, you have to base the model selection tests on the untransformed residuals.

•          One thing you should definitely not do is base your choice of weights (when you have more than one reasonable auxiliary equation) on the model selection tests. Here you are not selecting from different models, but choosing from different estimates (because of different weights) of the same model. Unfortunately, the weights enter into the SSE of the weighted residuals, and therefore into the model selection test statistics. Using different weights removes any basis for comparison; you are comparing apples and oranges. This is most obvious when the estimates are quite similar, but the weights are quite different. The model selection test statistics may be quite different (because the weights are different), but they should not be if they are to be used to select from alternative estimates, since the estimates are essentially the same.

•          Some continued to use an auxiliary equation that produced a large number of negative predicted values for the variances, replacing them with actual values. This is probably a bad idea. If a specification for the auxiliary equation produces a large number of negative predicated values, this should be treated as an indication that the specification is probably a misspecification, and the implied weights represent no improvement over OLS.

•          While the DIAG/HET command produces heteroskedasticity tests, these are not all based on NR² test statistics. However, when they are not (e.g., SSR tests such as those labelled as Glesjer and Harvey tests), they are based on the assumption that the disturbances are normally distributed. This additional information provides for a more powerful test – provided that the assumption is valid. On the other hand, the NR² test statistic is asymptotically chi-square no matter how the disturbances are distributed (provided the second moments are finite), so while these may be less powerful they are also of more general applicability. Unless you are confident that the disturbances are normally distributed, you should rely on the LM tests based on the NR² test statistic, eighter as reported by DIAG/HET or computed manually.

February 12, 2008