Consider two alternative regression models Y X0 11 e1 (9.21) E[X1e1] 0 Y X0 22 e2 (9.22) E[X2e2] 0 where X1

Consider two alternative regression models Y Æ X0 1¯1 Åe1 (9.21)

E[X1e1] Æ 0 Y Æ X0 2¯2 Åe2 (9.22)

E[X2e2] Æ 0 where X1 and X2 have at least some different regressors. (For example, (9.21) is a wage regression on geographic variables and (2) is a wage regression on personal appearance measurements.) You want to know if model (9.21) or model (9.22) fits the data better. Define ¾21

Æ E

£

e2 1

¤

and ¾22

Æ E

£

e2 2

¤

. You decide that themodel with the smaller variance fit (e.g., model (9.21) fits better if ¾21

Ç ¾22

.) You decide to test for this by testing the hypothesis of equal fit H0 : ¾21

Æ ¾22 against the alternative of unequal fit H1 : ¾21 6Æ ¾22

.

For simplicity, suppose that e1i and e2i are observed.

(a) Construct an estimator bµ of µ Æ ¾21

¡¾22

.

(b) Find the asymptotic distribution of p

n

¡bµ¡µ

¢

as n!1.

(c) Find an estimator of the asymptotic variance of bµ.

(d) Propose a test of asymptotic size ® of H0 against H1.

(e) Suppose the test accepts H0. Briefly, what is your interpretation?