Consider two alternative regression models

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?

Y = XB + e E[X1e1]=0 (9.21) Y = X22+ ez E[X2e2] = 0 (9.22)