Assume that the model y = X + u satisfies the Gauss-Markov assumptions, let be the

Assume that the model y = Xβ + u satisfies the Gauss-Markov assumptions, let β̂ be the OLS estimator of β. Let Z = G(X) be an n × (k + 1) matrix function of X and assume that Z’X[a(k + 1) × (k + 1) matrix] is nonsingular. Define a new estimator of β by β̃ = (Z’X)^–1Z’y.

(i) Show that E(β̃|X) = β, so that β̃ | is also unbiased conditional on X.

(ii) Find Var (β̃ | X). Make sure this is a symmetric, (k + 1) × (k + 1) matrix that depends on Z, X, and s².

(iii) Which estimator do you prefer, β̂ or β̃? Explain.