4. Relation between F and R. In our notes, we claim that F has an alternative expression (SSRR - SSRU)/#r F SSRU/(n - K)

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4. Relation between F and R. In our notes, we claim that F has an alternative expression (SSRR - SSRU)/#r F SSRU/(n - K) 1 ermitted where SSRR is the sum squared residuals from the restricted regression and SSRU is the sum of squared residuals from the unrestricted OLS. Take this as given to answer the following questions. Consider a regression that includes an intercept and X as regressors. Suppose the coeffi- cient on X is and the intercept is a, i.e., Yi = a + xi + . We want to test Ho: 8-0 vs H: 8+0. Suppose the dimension of is K - 1. (a) Under the restriction that = 0, what is the restricted least-squares estimator for a? (b) Show that SSRR - ( ), where i is the sample mean of y;. - (c) Use the result in (b) to show that (SSRR - SSRU) (i ), where ; is the fitted y; based on the unrestricted estimators for a and B. (Hint: first show that the cross-product term in the following expansion is zero.) n n n n n (Yi ) = (Yi i + i ) = (Yi ) + (i ) + 2(Yi 9i) (i ). - 2 i - (d) Use (0.1) and the results in part (c) to show that the F statistic for testing Ho (under conditional homoskedasticity) is given by R/(K-1) F (1 - R)/(n - K) (e) Use the result in (d) to show that nR d x(K 1) under Ho if conditional homoskedas- ticity holds. (f) Explain in words why nR can or cannot be used when the error is conditionally het- eroskedastic.