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Econometrics-I-23

Μ σ ir drawing u by random sampling eg requires

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μ σ ir Drawing u  by 'random sampling'   E.g.,      Requires many draws,  typically  hundreds or thousands
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Part 23: Simulation Based Estimation The Simulated Log Likelihood ™    11/25 ( 29 0 1 1 log log , ( ) ,..., i i1 iR where v  is the normally distributed effect.   Use the law of large numbers: let v v a random sample of R draws from  the standard normal po = = -∞ = + σ φ = T N it it i i i i t L g y v v dv x β ( 29 ( 29 0 0 1 1 1 , , ( ) pulation. 1 R = = = -∞ + σ → + σ φ T T R P it it iR it it i i i r t t g y v g y v v dv x x β β
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Part 23: Simulation Based Estimation Quasi-Monte Carlo Integration Based on Halton Sequences ™    12/25 0 1 0 ( ) I i i i I i r i i p = r b p H p b p = - - = = = Coverage of the unit interval is the objective, not randomness of the set of draws. Halton sequences --- Markov chain a prime number,               For example, using  × × × 0 1 2 -1 -2 -3 37 base p = 5, the integer r = 37 has b  = 2, b  = 2, b  = 1.  Then H (5) = 2 5  + 2 5  + 1 5  = 0.448.
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Part 23: Simulation Based Estimation Panel Data Estimation A Random Effects Probit Model ™    13/25 2 1 2 2 2 2 , 1,..., , 1,..., , ( 0), (observation mechanism) , ,..., ] ~ [ , ], ~ [0, ] ( , ) 1 1 [...] (1 ) , 1 1 it it it i it it i i iT i it it y u t T i N y y N u N Var = + ε + = = = ε ε σ ε σ = + σ = + σ x 1 0 x K L M O M K β Ι ρ ρ ρ ρ ρ ρ ρ ρ
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Part 23: Simulation Based Estimation Log Likelihood ™    14/25 1 1 2 2 1 1 1 h h log ( , ) log [(2 1)( )] ( ) = 1+ Quadrature log ( , ) log [(2 1)( )] W quadrature weight, z = quadrature = = -∞ = = = β σ = Φ - + σ φ σ ρ σ β σ ≈ Φ - + σ = T n it it i i i i t T n H h it it h i h t L y v v dv L W y z β x β x 1 1 1 ir i1 iR node Simulated 1 ˆ log ( , ) log [(2 1)( )] ˆ v = rth draw from standard normal for individual i.
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