525 1 i t i it i i i t u l g y v h v dv 1 1 1 2 1 φ

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™    5/25 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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Part 23: Simulation Based Estimation Obtaining the Unconditional Likelihood p 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
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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
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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
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Part 23: Simulation Based Estimation Butler and Moffitt’s Approach Random Effects Log Likelihood Function ™    9/25 ( 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 Butler and Moffitt: Compute this by Hermite quadrature  when  normal density  = quadrature node;  quadrature weight  =  is estimated with
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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
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