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Econometrics-I-22

Econometrics-I-22 - Applied Econometrics William Greene...

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Applied Econometrics William Greene Department of Economics Stern School of Business
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Applied Econometrics 22. Simulation Based Estimation
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Settings Conditional and unconditional log likelihoods Likelihood function to be maximized contains  unobservables Integration techniques Bayesian estimation Prior times likelihood is intractible How to obtain posterior means, which are open form  integrals The problem in both cases is “…how to do the  integration?”
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A Conditional Log Likelihood i i i i i i i i i i i v n i i=1 Conditional (on random v) density,                f(y  | , x , v ) Unconditional density:              f(y  | ,   x ) f(y  | , x , v )h(v | )dv Log likelihood function       log-L( ,  )= log f(y  | ,  θ θ α = θ α θ α θ i i i i i v x , v )h(v | )dv Integral does not exist in closed form.  How to do the maximization? α
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Application - Innovation Sample = 1,270 German Manufacturing Firms Panel, 5 years, 1984-1988 Response: Process or product innovation in the survey  year? (yes or no) Inputs:  Imports of products in the industry Pressure from foreign direct investment Other covariates Model:  Probit with common firm effects (Irene Bertschuk, doctoral thesis, Journal of  Econometrics, 1998)
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Likelihood Function Joint conditional (on u i ) density for obs. i. Unconditional likelihood for observation i How do we do the integration to get rid of the  heterogeneity in the conditional likelihood? 1 ( | ) ( ) = = i T i it i i i t u L g y v h v dv 1 1 1 ( ,..., | ) ( | ) [(2 1)( )] = = = = Φ - + σ T T i iT i it i it it i t t f y y v g y v y v β x
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Obtaining the Unconditional Likelihood The Butler and Moffitt (1982)  method is used by most current  software Quadrature (Stata –GLAMM) Works only for normally distributed  heterogeneity
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Hermite Quadrature H 2 h h h 1 2 1 2 f(x, v) exp( v )dv f(x, v )W Adapt to integrating out a normal variable exp( (v / ) ) f(x) f(x, v) dv 2 Change the variable to z = (1/( 2))v,              v =  ( 2)z and  , dv= ( 2)dz 1 f(x) f(x = -∞ -∞ -∞ - - σ = σ π σ σ σ = π 2 H h h=1 , z) exp( z )dz,   = 2 This can be accurately approximated by Hermite quadrature f(x) f(x, z)W λ - λ σ λ
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Example: 8 Point Quadrature Weights for 8 point Hermite Quadrature         0.661147012558199960,        0.20780232581489999,       0.0170779830074100010,        0.000199604072211400010  Nodes for 8 point Hermite Quadrature    Use both signs, + and -        0.381186990207322000,         1.15719371244677990        1.98165675669584300         2.93063742025714410
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Butler and Moffitt’s Approach Random Effects Log Likelihood Function ( 29 0 1 1 log log , ( ) = = -∞ = + T N it it i i i i t L g y v h v dv x β 1 0 ) ( ) ( ) h( ) , H i i i h h i h - h h i i f(v h v dv f z w v z w z v = = = σ σ β
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