(Hausman Test) The variance matrix difference (22.26) excited many people when it was first published by Hausman because it made the computation of the variance estimator for a difference in estimators a convenient by-product of the calculations of the two estimators.

(a) What problems can you anticipate with this variance estimator?

(b) Confirm the variance formula $Var[\delta_1 - \delta_2] = Var[\delta_2] - Var[\delta_1]$ in the following cases:

i. $\delta_1 = \beta_R$ and $\delta_2 = \beta$.

ii. $\delta_1 = \delta_{OLS}$ and $\delta_2 = \delta_{IV}$.

iii. $\delta_1 = \beta_{GLS}$ and $\delta_2 = \beta_{OLS}$.

(c) Show that the exogeneity test can also be interpreted as a test of whether the IV residuals are correlated with the residuals of the questionable explanatory variables after they have been regressed on the valid instruments.