Econometrics-I-21 - Econometrics I Professor William Greene...

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Part 21: Generalized Method of Moments Econometrics I Professor William Greene Stern School of Business Department of Economics
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Part 21: Generalized Method of Moments Econometrics I Part 21 – Generalized Method of Moments ™  1/61
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Part 21: Generalized Method of Moments The Method of Moments ™  2/61 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 = = μ = = θ θ θ = Σ Σ 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 ˆ g (m ,...,m ), k = 1,...,K = = μ = θ θ θ = Σ = θ θ θ θ =
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Part 21: Generalized Method of Moments Estimating a Parameter p Mean of Poisson n p(y)=exp(-λ) λy / y! n E[y]= λ. plim (1/N)Σiyi = λ. This is the estimator p Mean of Exponential n p(y) = λ exp(- λy) n E[y] = 1/ λ. plim (1/N)Σiyi = 1/λ ™  3/61
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Part 21: Generalized Method of Moments Mean and Variance of a Normal Distribution ™  4/61 = = = = - μ = - ÷ σ σ π = μ = σ + μ Σ = μ Σ = σ + μ μ σ = Σ - = Σ - 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 ˆ=y, ˆ y (y ) (y y)
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Part 21: Generalized Method of Moments Gamma Distribution ™  5/61 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. - λ = Γ = λ + = λ λ = - = Ψ - λ Ψ Γ E[y] and E[logy]. For a different course .... )
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Part 21: Generalized Method of Moments The Linear Regression Model ™  6/61 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 ] 0 Moment Equations 1 (y x x ... x )x 0 N 1 (y x x ... x )x 0 N ... 1 (y x x ... x )x 0 N Solution : Linea = = = = β + ε ε = - β - β - - β = - β - β - - β = - β - β - - β = r system of K equations in K unknowns. Least Squares
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Part 21: Generalized Method of Moments Instrumental Variables ™  7/61 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 0 N 1 (y x x ... x )z 0 N ... 1 (y x N = = = = β + ε ε = - β - β - - β = - β - β - - β = - β N i2 2 iK K iK -1 IV x ... x )z 0 Solution : Also a linear system of K equations in K unknowns. b = ( ) - β - - β = Z'X Z'y /n ( /n)
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Part 21: Generalized Method of Moments Maximum Likelihood ™  8/61 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 0 N Solution : K nonlinear equations in K un = = Σ θ θ = ∂θ θ θ = ∂θ N i i 1,MLE K,MLE i 1 k,MLE knowns.
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