Restricted Least Squares. (a) Show that RLS given by (7.36) is biased unless R = r. (b) Show that the var(RLS) = var(A(XX)1Xu) where A

Restricted Least Squares.

(a) Show that βRLS given by (7.36) is biased unless Rβ = r.

(b) Show that the var(βRLS) = var(A(XX)−1Xu) where A = IK − (X



X)−1R

[R(X



X)−1R

]−1R.

Prove that A2 = A, but A = A. Conclude that var(βRLS) = σ2A(X



X)−1A

 = σ2{(X



X)−1

−(X



X)−1R

[R(X



X)−1R

]−1R(X



X)−1}.

(c) Show that var(βOLS)− var(βRLS) is a positive semi-definite matrix.