Question:

(theoretical question, easy) Let 'V be the sample variance-covariance matrix of f, ' (FxF) (i.e., V= LS (B E)(f ), where f = L 5"1 f, (the sample mean of f,)). Define b = V! Show, in two or three lines, that (0] E 0 = |_ 11 . (11) is a consistent estimator of . You can take for granted that A is consistent for A, that the sample mean f is consistent for the population mean p, and that the o~ sample variance-covariance matrix V is consistent for the population variance-covariance matrix V = Var(f;). Hint: A = Var (fi) b. . To summarize the discussion in class, the risk premium model is E(Rit - Rot) = b' Cov(ft, Rat - Rot), i = 1, 2, ..., N. (1) Here, Rot is (one plus) the safe rate of return, f is the factor vector, F is the (Fx1) number of factors, N is the number of "test assets". The left hand side is asset i's risk premium. As shown in class, this model can be translated into the following zero-mean conditions: E[(Rit - Rot) - (Rit - Rot)b'(ft - ()] =0, i = 1, 2, ..., N, (2) E(ft - 1) = 0 (3) ( FX1)' where = E(f,). Therefore, the number of the zero-mean condition is N + F. The parameter vector o is b 6 (Fx1) (2Fx1) LL (Fx1) . The GMM setup. To get GMM rolling, all you need to do is to find the g funciton. we = (r, , ft). (4) g1 (Wt; 6) g(wt; 6) = (Nx1) , gi( w; 6) = re - re (f - u)' b , g2(wa;6) =fi-A, ((N+F)x1) g2 (Wt; 6) (Nx1) (Nx1) (Nx1) (1xF) (Fx1)' (Fx1) (Fx1) (5) Rit - Rot Rat - Rot r (N-vector of excess returns). (6) (Nx1) RNt - Rot]You should verify that, if g(w; 6) is defined this way, then the N + F zero-mean conditions are Elg(wt; 6)] = 0 (where o is the true parameter vector). The ((N+F) x1) rest is automatic: gn (6) - (7) ((N+F)x1) n 1=1 (GMM estimate) 6(W) = argmin gn (6)'Wg (6). (8) . In what follows, assume that { Rit - Rot, ..., RNt - Rot, fo} is ergodic stationary and {g(w; 6)} is mds (a martingale difference sequence). By Billingsley's CLT, 1 [g(w; 6) + NO, S ), S = E(gig.), g. = g(Wt; 6). (9) (N+F) x (N+F) t=1(theoretical question) Let (b, () be the GMM estimator. Go back to the linear factor model (1). Show, in two or three lines, that a consistent estimator of the right hand side b' Cov(ft, Rit - Rot) is b'- (ft - p) (Rit - Rot). (15) 1=1 Hint: -) (ft - p)(Rat - Rot) = E fe( Rat - Rot) - H- (Rat - Rot)- t=1 t=1 t=1 Write the covariance in terms of expectations. That is, with x = f, and y = Rit - Rot, Cov(x, y) = E(x . y) - E(x) E(y).The "beta" representation of the linear factor model is E(Rit - Rot) = 3; x with Bi E Var (fi) Cov (ft, Rit - Rot), X = Var (ft) b (1xF)(Fx1) (Fx1) (FxF) (Fx1) (Fx1) (FxF) (Fx1) (10)(f) (theoretical question, a matrix-based expression of gn (6), very easy) Show that the gn (6) defined in (7) can be written as 1 + u'b re - Mref b (N x1) gn (6) = (N xF)(Fx1) (12) ((N+F)x1) f - 1 (Fx1) (Fx1) where re re Mref f, = ft (13) (Nx1) t=1 (Nx1) (N XF) n t=1 (Nx1) (1XF) (Fx1) t=1 (Fx1)