Just-Identified Model. Consider the just-identified equation y1 = Z11 + u1 with W, the matrix of instruments for this equation of dimension T

Just-Identified Model. Consider the just-identified equation y1 = Z1δ1 + u1 with W, the matrix of instruments for this equation of dimension T ×  where  = g1 + k1 the dimension of Z1. In this case, WZ1 is square and nonsingular.

(a) Show that the generalized instrumental variable estimator given below (11.41) reduces to the simple instrumental variable estimator given in (11.38).

(b) Show that the minimized value of the criterion function for this just-identified model is zero, i.e., show that (y1 − Z1δ1,IV )PW(y1 − Z1δ1,IV) = 0.

(c) Conclude that the residual sum of squares of the second stage regression of this just-identified model is the same as that obtained by regressing y1 on the matrix of instruments W, i.e., show that (y1 −  Z1δ1,IV )(y1 −  Z1δ 1,IV) = y

1

¯ PWy1 where  Z1 = PWZ1. Hint: Show that P

Z1

= PPWZ1 = PW, under just-identification.