Let e be the GLS estimator (4.22) under the assumptions (4.18) and (4.19). Assume that is known and 2 is fdunknown. Define the residual

Let e¯ be the GLS estimator (4.22) under the assumptions (4.18) and (4.19). Assume that §

is known and ¾2 is fdunknown. Define the residual vector e Æ Y ¡X e¯, and an estimator for ¾2 e¾

2 Æ

1 n ¡k e0§¡1ee.

(a) Show (4.23).

(b) Show (4.24).

(c) Prove that e ÆM1e, where M1 Æ I ¡X

¡

X 0§¡1X

¢¡1 X 0§¡1.

(d) Prove that M0 1§¡1M1 Æ §¡1 ¡§¡1X

¡

X 0§¡1X

¢¡1 X 0§¡1.

(e) Find E £
e¾
2 j X ¤
.
(f ) Is e¾
2 a reasonable estimator for ¾2?