3 3 140 points total if e h θ w t the gmm estimate

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–3– 3.) (140 points total) If E [ h ( θ 0 , w t )] = 0 , the GMM estimate of θ is de fi ned as the value that minimizes T [ g ( θ , Y T )] 0 S 1 [ g ( θ , Y T )] where g ( θ , Y T ) = T 1 T X t =1 h ( θ , w t ) S = lim T →∞ T 1 T X t =1 X v = −∞ E n [ h ( θ 0 , w t )] £ h ( θ 0 , w t v ) ¤ 0 o . The fi rst-order conditions for this minimization are · g ( θ , Y T ) θ 0 ¸ 0 S 1 g ( θ , Y T ) = 0 . The usual asymptotic distribution of the GMM estimator is T ( ˆ θ T θ 0 ) L N ( 0 , V ) where V = ( DS 1 D 0 ) 1 D 0 = E ( h ( θ , w t ) θ 0 ¯ ¯ ¯ ¯ θ = θ 0 ) . Hansen’s J -statistic is given by J = T [ g ( ˆ θ , Y T )] 0 ˆ S 1 [ g ( ˆ θ , Y T )] for ˆ S a consistent estimate of S .
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–4– In this question you are asked to apply these results to the two-stage least squares re- gression model, in which our task is to estimate the ( a × 1 ) vector θ where y t = z 0 t θ + u t using an ( r × 1 ) vector of instruments x t satisfying E ( x t u t ) = 0 . You can assume that r > a , that the vector ( y t , z 0 t , x 0 t ) is stationary and ergodic, that u t x t is a martingale di ff erence sequence, that E ( u 2 t | x t ) = σ 2 , that the matrix Σ xx = E ( x t x 0 t ) has rank r, and that the matrix Σ xz = E ( x t z 0 t ) has rank a. a.) (10 points) Are there any other assumptions besides those stated in the above para- graph that you would need to be able to apply the GMM results stated on page 3 to this particular example? b.) (30 points) Find the values for g ( θ , Y T ) , S , and D 0 for this example. c.) (20 points) Propose consistent estimates of S and D , denoted ˆ S and ˆ D , respectively. d.) (20 points) Show that for this example the GMM estimate ˆ θ GMM is equivalent to the 2SLS estimate ˆ θ 2 SLS = h ( Σ z t x 0 t ) ( Σ x t x 0 t ) 1 ( Σ x t z 0 t ) i 1 h ( Σ z t x 0
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