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Unformatted text preview: Applied Econometrics William Greene Department of Economics Stern School of Business Applied Econometrics 22. Simulation Based Estimation 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?” 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 logL( , )= 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 ∫ Application  Innovation Sample = 1,270 German Manufacturing Firms Panel, 5 years, 19841988 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) 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 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 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 ∑ 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 Butler and Moffitt’s Approach...
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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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