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Unformatted text preview: Applied Econometrics William Greene Department of Economics Stern School of Business Applied Econometrics 20. The Generalized Method of Moments The Method of Moments k k k 1 2 K N k N 1 1 k i 1 i k i 1 k i N N Estimating Parameters of Distributions Using Moment Equations Population Moment E[x ] f ( , ,..., ) Sample Moment m x  m may also be h (x ), need not be powers Law of L = = = = = k k k 1 2 K N k 1 k i 1 i k 1 2 K N k k 1 K arge Numbers plim m f ( , ,..., ) 'Moment Equation' (k = 1,...,K) m x f ( , ,..., ) Method of Moments g (m ,...,m ), k = 1,...,K = = = = = = Estimating a Parameter Mean of Poisson p(y)=exp( ) y / y! E[y]= . plim (1/N) i y i = . This is the estimator Mean of Exponential p(y) = exp( y) E[y] = 1/ . plim (1/N) i y i = 1/ Mean and Variance of a Normal Distribution 2 2 2 2 2 N N 2 2 2 1 1 i 1 i i 1 i N N 2 N 2 2 N 2 1 1 i 1 i i 1 i N n 1 (y ) p(y) exp 2 2 Population Moments E[y] , E[y ] Moment Equations y , y Method of Moments Estimators =y, y (y ) (y y) = = = =  = = = + = = + =  =  Gamma Distribution P P 1 2 2 exp( y)y p(y) (P) P E[y] P(P 1) E[y ] E[1/ y] P 1 E[logy] (P) log , (P)=dln (P)/dP (Each pair gives a different answer. Is there a 'best' pair? Yes, the ones that are 'sufficient' statistics.   = = + = = =  E[y] and E[logy]. For a different course....) The Linear Regression Model i i i i ik N i i1 1 i2 2 iK K i1 i 1 N i i1 1 i2 2 iK K i2 i 1 N i i1 1 i2 2 iK K iK i 1 Population y x Population Expectation E[ x ] Moment Equations 1 (y x x ... x )x N 1 (y x x ... x )x N ... 1 (y x x ... x )x N Solution: Linea = = = = + =   =   =   = r system of K equations in K unknowns. Least Squares Instrumental Variables i i i i ik 1 K N i i1 1 i2 2 iK K i1 i 1 N i i1 1 i2 2 iK K i2 i 1 i i1 1 i 1 Population y x Population Expectation E[ z ] 0 for instrumental variables z ... z . Moment Equations 1 (y x x ... x )z N 1 (y x x ... x )z N ... 1 (y x N = = = = + =   =   = N i2 2 iK K iK1 IV x ... x )z Solution: Also a linear system of K equations in K unknowns. b = ( )  = Z'X Z'y /n ( /n) Maximum Likelihood N 1 i 1 i i 1 K N k N i i 1 K i 1 k Log likelihood function, logL = logf(y  x , ,..., ) Population Expectations logL E 0, k = 1,...,K Sample Moments logf(y  x , ,..., ) 1 N Solution : K nonlinear equations in K un = = = = N i i 1,MLE K,MLE i 1 k,MLE knowns....
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This note was uploaded on 11/23/2011 for the course ECON B30.3351 taught by Professor Professorw.greene during the Spring '10 term at NYU.
 Spring '10
 ProfessorW.Greene
 Econometrics

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