Take a linear equation with endogeneity and a just-identified linear reduced form Y Xe with X Z u2 where both X and Z

Take a linear equation with endogeneity and a just-identified linear reduced form Y Æ

X¯Åe with X Æ °Z Åu2 where both X and Z are scalar 1£1. Assume that E[Ze] Æ 0 and E[Zu2] Æ 0.

(a) Derive the reduced formequation Y Æ Z¸Åu1. Show that ¯ Æ ¸/° if ° 6Æ 0, and that E[Zu] Æ 0.

(b) Letb¸

denote the OLS estimate from linear regression of Y on Z, and let b° denote the OLS estimate from linear regression of X on Z. Write µ Æ (¸,°)0 and let bµ Æ (b¸, b°)0. Define u Æ (u1,u2). Write p n

¡bµ¡µ

¢

using a single expression as a function of the error u.

(c) Show that E[Zu] Æ 0.

(d) Derive the joint asymptotic distribution of p

n

¡bµ¡µ

¢

as n!1. Hint: Define ­u Æ E

£

Z2uu0¤

.

(e) Using the previous result and the Delta Method find the asymptotic distribution of the Indirect Least Squares estimator b¯ Æb¸/b°.

(f ) Is the answer in

(e) the same as the asymptotic distribution of the 2SLS estimator in Theorem 12.2?

Hint: Show that

¡

1 ¡¯

¢

u Æ e and

¡

1 ¡¯

¢

­u

µ

1

¡¯

¶

Æ E

£

Z2e2¤

.