(Exact Multicollinearity) Consider a situation in which Assumptions 6.1

(First Moments, p. 110) and 7.1

(Second Moments, p. 130) hold but Assumption 3.1

(Full Rank, p. 53) is violated so that X is rank deficient.

(a) What can we infer about the variance matrix of $\hat{\beta}$ under such conditions?

(b) How does exact multicollinearity affect $Var[\hat{\mu}|X]$?

(c) Show that $E[(\hat{\mu} - \mu_0)(\hat{\mu} - \mu_0)/rank(X'|X)] = \sigma^2$. Find an unbiased estimator for $\sigma^2$

allowing for exact multicollinearity.