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Econ226_IF

# Econ226_IF - I Bayesian econometrics A Introduction B...

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1 I. Bayesian econometrics A. Introduction B. Bayesian inference in the univariate regression model C. Statistical decision theory D. Large sample results E. Diffuse priors F. Numerical Bayesian methods 1. Importance sampling Generic Bayesian problem: p Y | 2   ± likelihood (known) p 2   ± prior (known) goal: calculate p 2 | Y   ± p Y | 2   p 2   G for G ± ; p Y | 2   p 2   d 2 Analytical approach: choose p 2   from a family such that G can be found with clever algebra. Numerical approach: satisfied to be able to generate draws 2 1   , 2 2   ,..., 2 D   from the distribution p 2 | Y   without ever knowing the distribution (i.e., without calculating G )

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2 Importance sampling: Step (1): Generate 2 j   from an (essentially arbitrary) “importance density” g 2   . Step (2): Calculate F j   ± p ¡ Y | 2 j   ¢ p ¡ 2 j   ¢ g ¡ 2 j   ¢ . Step (3): Weight the draw 2 j   by F j   to simulate distribution of p 2 | Y   . Examples: E 2 | Y   ± ; 2 p 2 | Y   d 2 S ! j ± 1 D 2 j   F j   ! j ± 1 D F j   q 2 ' Var 2 | Y   S ! j ± 1 D 2 j   " 2 '   2 j   " 2 '   U F j   ! j ± 1 D F j
3 Prob 2 2 ² 0   S ! j ± 1 D - ¡ 2 2 j   ² 0 ¢ F j   ! j ± 1 D F j   How does this work? ! j ± 1 D 2 j   F j   ! j ± 1 D F j   ± D " 1 ! j ± 1 D 2 j   F j   D " 1 ! j ± 1 D F j   Numerator: D " 1 ! j ± 1 D 2 j   F j   p v E ¡ 2 j   F j   ¢ ± ; 2 F 2   g 2   d 2 ± ; 2 p Y | 2   p 2   g 2   g 2   d 2 ± ; 2 p Y | 2   p 2   d 2

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4 Denominator: D " 1 !
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