Question: Take themodel Y 0e E[X j Z] 0Z E[e j Z] 0 where Y is scalar, X is a k

Take themodel Y Æ ¼0¯Åe

¼ Æ E[X j Z] Æ ¡0Z E[e j Z] Æ 0 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.

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