Econometrics-I-23

The butler and moffitt(1982 method is used by most

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Unformatted text preview: The Butler and Moffitt (1982) method is used by most current software n Quadrature (Stata –GLAMM) n Works only for normally distributed heterogeneity ˜™™ ™ 6/25 Part 23: Simulation Based Estimation Hermite Quadrature ˜™™ ™ 7/25 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 ∑ Part 23: Simulation Based Estimation 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 ˜™™ ™ 8/25 Part 23: Simulation Based Estimation Butler and Moffitt’s Approach Random Effects Log Likelihood Function ˜˜™™ ™ 9/25 ( 29 1 1 log log , ( ) ∞ = =-∞ ′ = + ∑ ∏ ∫ T N it it i i i i t L g y v h v dv x β 1 ) ( ) ( ) 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 Butler and Moffitt: Compute this by Hermite quadrature when normal density = quadrature node; quadrature weight = is estimated with Part 23: Simulation Based Estimation Monte Carlo Integration ˜˜™™ ™ 10/25 1 1 ( ) ( ) ( ) [ ( )] i i R P ir i i i u i u r f u f u g u du E f u R = → = ∑ ∫ (Certain smoothness conditions must be met.) 1 2 ( ), ~ [0,1] ( ) [ , ] ir ir ir ir ir u t v v U u v for N- = = σΦ + μ μ σ ir Drawing u by 'random sampling' E.g., Requires many draws, typically hundreds or thousands Part 23: Simulation Based Estimation The Simulated Log Likelihood ˜˜™™ ™ 11/25 ( 29 1 1 log log , ( )...
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The Butler and Moffitt(1982 method is used by most current...

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