Let be the OLS estimator in (1), where the design matrix X has full rank p

Let βˆ be the OLS estimator in (1), where the design matrix X has full rank p

(a) Show that E(Y |X) = Xβ and cov(Y |X) = σ2In×n. Verify that E(βˆ|X) = β and cov(βˆ|X) = σ2(X

X)−1.

(b) Suppose c is p×1. Show that E(c

βˆ|X) = c

β and var(c

βˆ|X) =

σ2c

(X

X)−1c.

Hint: look at the proofs of theorems 4.2 and 4.3. Bigger hint: look at equations (4.8–10).