There are different ways to combine features of the Breusch-Pagan and White tests for heteroskedasticity. One possibility not covered in the text is to run the regression
on xi1, xi2, ...., xik, ŷ2i, i = 1,...., n,
where the u. are the OLS residuals and the ŷi. are the OLS fitted values. Then, we would test joint significance of xi1, xi2, ....., xik and ŷ2i. (Of course, we always include an intercept in this regression.)
(i) What are the df associated with the proposed F test for heteroskedasticity?
(ii) Explain why the R-squared from the regression above will always be at least as large as the R-squareds for the BP regression and the special case of the White test.
(iii) Does part (ii) imply that the new test always delivers a smaller p-value than cither the BP or special case of the White statistic? Explain.
(iv) Suppose someone suggests also adding ŷi to the newly proposed test. What do you think of this idea?