In the linear model Y X0e with E[Xe] 0 the GMMcriterion function for is J () 1 n Y X

In the linear model Y Æ X0¯Åe with E[Xe] Æ 0 the GMMcriterion function for ¯ is J (¯) Æ

1 n

¡

Y ¡X ¯

¢0 Xb­

¡1X 0 ¡

Y ¡X ¯

¢

(13.29)

where b­

Æ n¡1Pni

Æ1 Xi X0 i be2 i , bei Æ Yi ¡ X0 i

b¯ are the OLS residuals, and b¯ Æ

¡

X 0X

¢¡1 X 0Y is least squares.

The GMMestimator of ¯ subject to the restriction r (¯) Æ 0 is e¯ Æ argmin r (¯)Æ0 Jn(¯).

The GMMtest statistic (the distance statistic) of the hypothesis r (¯) Æ 0 is D Æ J ( e¯) Æ min r (¯)Æ0 J (¯). (13.30)

(a) Show that you can rewrite J (¯) in (13.29) as J (¯) Æ n

¡

¯¡ b¯

¢0 bV ¡1

¯

¡

¯¡ b¯

¢

and thus e¯ is the same as the minimumdistance estimator.

(b) Show that under linear hypotheses the distance statistic D in (13.30) equals theWald statistic.