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Hypothesis Tests of ML and GMM Estimation

# Hypothesis Tests of ML and GMM Estimation - ECON 203C...

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Unformatted text preview: ECON 203C: System Models TA Note 6: Version 1 Hypothesis Tests of ML and GMM Estimations Hisayuki Yoshimoto Last Modi&ed: May 20, 2008 Abstract: 1 Wald, Lagrange Multiplier (LM), and Likelihood Ratio (LR) Statistics 1.1 Review of ML and GMM estimator Assume that f w i g N i =1 = f y i ;x i g N i =1 are i.i.d. samples. We have the nonlinear model (which might be linear) y i = g ( x i ;& ) + u i ; where & is K & 1 vector of parameters. 1.1.1 ML Estimator (Review) Assume the error term u i has conditional pdf f ( y i j x i ; & ) : Then, the conditional likelihood is de&ned as L ( & ) = N Q i =1 f ( y i j x i ; & ) ; and log conditional likelihood function is l ( & ) = ln N Q i =1 f ( y i j x i ; & ) = N P i =1 ln f ( y i j x i ; & ) : De&ne the objective function Q n ( & ) such that Q n ( & ) = 1 N N P i =1 ln f ( y i j x i ; & ) Then, ML estimator is de&ned as the solution of f.o.c. equation @ @& Q n ( & ) = @ @& 1 N N P i =1 ln f ( y i j x i ; & ) = 1 N N P i =1 @ @& ln f ( y i j x i ; & ) | {z } score = 0 K & 1 : Also, the asymptotic distribution of ML estimator is p N & ^ & ML ¡ & ¡ d ! N @ K & 1 ; & |{z} K & K 1 A ; where I 1 is the Fisher information & |{z} K & K = I ¡ 1 1 = ¡ E ¢ @ @&@& ln f ( y i j x i ; & ) £ : 1 We estimate I 1 by ^ I 1 = & 1 N N P i =1 @ @&@& ln f ( y i j x i ; & ) : Thus, estimate of & is ^ & |{z} K & K = ^ I ¡ 1 1 = & & 1 N N P i =1 @ @&@& ln f ( y i j x i ; & ) ¡ ¡ 1 Thus, we have the approximated distribution p N ¢ ^ & ML & & £ A ¡ N B @ K & 1 ; ^ I ¡ 1 1 |{z} K & K 1 C A or equivalently, ^ & ML A ¡ N B B @ &; 1 N ^ I ¡ 1 1 | {z } K & K 1 C C A 1.1.2 Optimal GMM Estimator (Review) Assume that we have M ¢ 1 dimensional moment condition E 2 6 4 ’ ( w i ;& ) | {z } M & 1 3 7 5 = 0 M & 1 : (moment condition) The candidate of moment condition is E 2 6 4 u i £ h ( x i ) | {z } M & 1 3 7 5 = E [( y i & g ( x i )) £ h ( x i )] = 0 M & 1 ; where h ( x i ) is any continuous and di/erentiable M ¢ 1 dimensional function. Keep discussing with general notation of moment function ’ ( w i ;& ) : Sample analogue is m n ( & ) | {z } M & 1 = 1 N N P i =1 ’ ( w i ;& ) | {z } M & 1 = 0 M & 1 (sample analogue) De&ne the objective function Q n ( & ) such that Q n ( & ) | {z...
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Hypothesis Tests of ML and GMM Estimation - ECON 203C...

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