3. (#6) Take the linear model yi X' B+ es E (zie) = 0 and consider the GMM estimator of 8. Let Jn (B) =n.in =non (8)'*n (8) denote the test of overidentifying restrictions. Show that In (B) -axi-k as n by proving each of the following =n. (a) Since 2 > 0, we can write 2-1 = CC and N = (C)-40-1 (Hint: Since 2-1 is symmetric, there exists a matrix H such that 1-1 =HAH' where A is a diagonal matrix with positive values and HH' =I) (b) In (8) (C3, (8)) (cc) C3 (8) (c) Con (B) = DCin (Bo) where Dn=11-C [CE* xz) 0672 ZX (2x)]*(+xzba'(c)-* n and n In (B) = z'e. (Hint: Recall that 8 80 = [(4 x'z) (47'x)]- (4x'z) 1 1 2 Z' e = [(4x'z)a * (+2' x)]** (1 x'zbn "In (Bo) we can write In (8) IZ' (Y-X (8 Bo)] x3) -Z' Y X3 -X ) (d) Dr -, 11-R(RR)-R' where R= C'E (Z;x) (Hint: Use the fact that 12) (e) VC 9, (30) (0,1) (f) In () - ~ N(0,1,Y (1, - R (R'R) R] N (0,19). (Use the results of (b) through (e)) (3) N (0,1,Y (11 R (RR) R) 1 (0,1,) = xi_k (Hint: When M = 1; R (RR) - R we can write M = A' AMA where Il-k AM 0 Ole and AA' = 1) 3. (#6) Take the linear model yi X' B+ es E (zie) = 0 and consider the GMM estimator of 8. Let Jn (B) =n.in =non (8)'*n (8) denote the test of overidentifying restrictions. Show that In (B) -axi-k as n by proving each of the following =n. (a) Since 2 > 0, we can write 2-1 = CC and N = (C)-40-1 (Hint: Since 2-1 is symmetric, there exists a matrix H such that 1-1 =HAH' where A is a diagonal matrix with positive values and HH' =I) (b) In (8) (C3, (8)) (cc) C3 (8) (c) Con (B) = DCin (Bo) where Dn=11-C [CE* xz) 0672 ZX (2x)]*(+xzba'(c)-* n and n In (B) = z'e. (Hint: Recall that 8 80 = [(4 x'z) (47'x)]- (4x'z) 1 1 2 Z' e = [(4x'z)a * (+2' x)]** (1 x'zbn "In (Bo) we can write In (8) IZ' (Y-X (8 Bo)] x3) -Z' Y X3 -X ) (d) Dr -, 11-R(RR)-R' where R= C'E (Z;x) (Hint: Use the fact that 12) (e) VC 9, (30) (0,1) (f) In () - ~ N(0,1,Y (1, - R (R'R) R] N (0,19). (Use the results of (b) through (e)) (3) N (0,1,Y (11 R (RR) R) 1 (0,1,) = xi_k (Hint: When M = 1; R (RR) - R we can write M = A' AMA where Il-k AM 0 Ole and AA' = 1)