By William Greene

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The limiting distribution of v′v is chi-squared J if the limiting distribution of v is standard normal. All of the conditions for the central limit theorem apply to v, so we do have the result we need. This implies that as long as the data are well behaved, the numerator of the F statistic will converge to the ratio of a chi-squared variable to its degrees of freedom. 4. Finally, suppose that Ω must be estimated, but that assumptions (10-27) and (10-31) are met by the estimator. What changes are required in the development of the previous problem?

To carry out the Goldfeld-Quandt test, we order the data first based on X1 then on X2. The regressions are computed using the first and last 17 observations, so the F statistic in each case is F[14,14] = e1′e1 / e2′e2 where e1′e1 is the larger of the two sums of squares and e2′e2 is the smaller. For our data set, we Sorted on X2 find Sorted on X1 e′e for obs. 026 e′e for obs. 48. We would conclude, therefore, that there is evidence of heteroscedasticity and it is related to X2 but not X1. In view of this finding, it is instructive to go back to the White and Breusch and Pagan tests considered earlier.

This is a surprising outcome. The likelihood ratio statistic is based on both models. The sum of squared residuals for the restricted model is given above. 44757. 61998. Once again, the statistic is small. 12. 5621. All of these suggest that the log-linear model is not a significant restriction on the Box-Cox model. This rather peculiar outcome would appear to arise because of the rather substantial reduction in the log-likelihood function which occurs when the dependent variable is transformed along with the right hand side.