(Hausman Test) Reconsider the Hausman specification test for the random-effects model (24.9)--(24.11).

(a) Construct a Hausman specification test statistic using the difference between the OLS and LSDV estimators.

(b) Show that this test statistic is identical to Hausman's test statistic under the random-effects specification (24.12) of the conditional variance of $$\textbf{y}_i$$. [Hint: Show first that $$\hat{\beta}_{OLS}(1) = \textbf{A}(1)\hat{\beta}_{DV} + [\textbf{I}_k - \textbf{A}(1)]\bar{\beta}$$, where $$\textbf{A}(.)$$ is defined in (24.22).]

(c) Generalize the test to cases in which the conditional variance of $$\textbf{y}_i$$ is heteroskedastic and not equicorrelated.

(d) Derive an asymptotically equivalent gradient test statistic.