Take themodel Y X0e with E[Xe] 0 where X contains an intercept so E[e] 0. An enterprising econometrician notices that this implies

Take themodel Y Æ X0¯Åe with E[Xe] Æ 0 where X contains an intercept so E[e] Æ 0. An enterprising econometrician notices that this implies the n moment conditions E[ei ] Æ 0, i Æ 1, ...,n.

Given an n £n weight matrixW, this implies a GMMcriterion J (¯) Æ

¡

Y ¡X ¯

¢0W

¡

Y ¡X ¯

¢

.

(a) Under i.i.d. sampling, show that the efficient weight matrix isW Æ ¾¡2I n where ¾2 Æ E

£

e2¤

.

(b) Using the weightmatrixW Æ ¾¡2I n find the GMMestimator b¯ that minimizes J (¯).

(c) Find a simple expression for the minimized criteria J ( b¯).

(d) Theorem 13.14 says that criterion such as J ( b¯) are asymptotically Â2

`¡k where ` is the number of moments. While the assumptions of Theorem 13.14 do not apply to this context, what is ` here?

That is, which Â2 distribution is the asserted asymptotic distribution?

(e) Does the answer in

(d) make sense? Explain your reasoning.