Take the model Y X0e with E[e j X] 0 and E e2 j X 2. An econometrician more enterprising

Take the model Y Æ X0¯Åe with E[e j X] Æ 0 and E

£

e2 j X

¤

Æ ¾2. An econometrician more enterprising than the one in previous question notices that this implies the nk moment conditions E[Xi ei ] Æ 0, i Æ 1, ...,n.

We can write the moments using matrix notation as E h

X 0 ¡

Y ¡X ¯

¢i where X Æ

0 BBBB@

X0 1 0 ¢ ¢ ¢ 0 0 X0 2 0

...

...

...

0 0 ¢ ¢ ¢ X0 n

1 CCCCA.

Given an nk £nk weight matrixW this implies a GMM criterion J (¯) Æ

¡

Y ¡X ¯

¢0 XW X 0 ¡

Y ¡X ¯

¢

.

(a) Calculate ­Æ E h

X 0

ee0X i

.

(b) The econometrician decides to set W Æ ­¡, the Moore-Penrose generalized inverse of ­. (See Section A.6.) Note: A useful fact is that for a vector a,

¡

aa0¢¡

Æ aa0 ¡

a0a

¢¡2 .

(c) Find the GMMestimator b¯ that minimizes J (¯).

(d) Find a simple expression for the minimized criterion J ( b¯).

(e) Comment on whether the Â2 approximation from Theorem 13.14 is appropriate for J ( b¯).