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Econometrics-I-23

I i1 ir where v is the normally distributed effect

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Unformatted text preview: ( ) ,..., 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 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 Part 23: Simulation Based Estimation Quasi-Monte Carlo Integration Based on Halton Sequences ˜˜™™ ™ 12/25 1 ( ) 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 × × × 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. 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 ε σ ⊥ ε σ = + σ = + σ x 1 x K L M O M K β Ι ρ ρ ρ ρ ρ ρ ρ ρ 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. ˆ ˆ (v ,...,v ) are reused for all co = = = ′ β σ ≈ Φ- + σ ∑ ∑ ∏ T n R it it ir i r t L y v R β x mputations of function or derivatives. Part 23: Simulation Based Estimation Application: Innovation ˜˜™ ™ 15/25 Part 23: Simulation Based Estimation Application: Innovation ˜˜™ ™ 16/25 Part 23: Simulation Based Estimation (1.17072 / (1 + 1.17072) = 0.578) ˜˜™ ™ 17/25 Part 23: Simulation Based Estimation...
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i i1 iR where v is the normally distributed effect Use the...

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