Suppose Y = X+. The design matrix X is np with rank p

Suppose Y = Xβ+. The design matrix X is n×p with rank p

  X. The i are independent with E(i) = 0. However, var(i) = λci.

The ci are known positive constants.

(a) If λ is known and the ci are all equal, show that the GLS estimator for β is the p×1 vector γ that minimizes



i (Yi − Xiγ )2.

(b) If λ is known, and the ci are not all equal, show that the GLS estimator for β is the p×1 vector γ that minimizes



i (Yi − Xiγ )2/var(Yi|X).

Hints: In this application, what is the ith row of the matrix equation

(9)? How is (9) estimated?

(c) If λ is unknown, show that the GLS estimator for β is the p × 1 vector γ that minimizes



i (Yi − Xiγ )2/ci.