Take themodel where Y is scalar, X is a k vector and Z is an ` vector. and are k 1 and

Take themodel

where Y is scalar, X is a k vector and Z is an ` vector. ¯ and ¼ are k £1 and ¡ is `£k. The sample is (Yi ,Xi ,Zi : i Æ 1, ...,n) with ¼i unobserved.
Consider the estimator b¯ for ¯ by OLS of Y onb¼
Æb¡
0Z whereb¡
is the OLS coefficient from the multivariate regression of X on Z.

(a) Show that b¯ is consistent for ¯.

(b) Find the asymptotic distribution p n ¡ b¯¡¯
¢
as n!1assuming that ¯ Æ 0.

(c) Why is the assumption ¯ Æ 0 an important simplifying condition in part (b)?

(d) Using the result in

(c) construct an appropriate asymptotic test for the hypothesis H0 : ¯ Æ 0.

Y = '+ =E[X]Z]='Z E[e]Z]=0