Question: Take the linear model Y X0e with E[Ze] 0. Consider the GMM estimator b of . Let J n g n( b)0b

Take the linear model Y Æ X0¯Åe with E[Ze] Æ 0. Consider the GMM estimator b¯ of ¯.

Let J Æ n g n( b¯)0b­

¡1g n( b¯) denote the test of overidentifying restrictions. Show that J ¡!

d

Â2

`¡k as n !1 by demonstrating each of the following.

(a) Since ­È 0, we can write ­¡1 ÆCC0 and ­ÆC0¡1C¡1 for some matrix C.

(b) J Æ n

¡

C0g n( b¯)

¢0 ¡

C0b­C

¢¡1C0g n( b¯).

(c) C0g n( b¯) ÆDnC0g n(¯) where g n(¯) Æ 1n Z0e and Dn Æ I ` ¡C0

µ

1 n

Z0X

¶µµ

1 n

X 0Z

b­

¡1

µ

1 n

Z0X

¶¶¡1 µ

1 n

X 0Z

b­

¡1C0¡1.

(d) Dn ¡!

p I ` ¡R

¡

R0R

¢¡1R0 where R ÆC0E

£

Z X0¤

.

(e) n1/2C0g n(¯) ¡!

d u »N(0, I `) .

(f ) J ¡!

d u0

³

I ` ¡R

¡

R0R

¢¡1R0

´

u.

(g) u0

³

I ` ¡R

¡

R0R

¢¡1R0

´

u » Â2

`¡k .

Hint: I ` ¡R

¡

R0R

¢¡1R0 is a projection matrix.

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