Econometrics-I-24 - Econometrics I Professor William Greene...

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Part 24: Bayesian Estimation Econometrics I Professor William Greene Stern School of Business Department of Economics
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Part 24: Bayesian Estimation Econometrics I Part 24 – Bayesian Estimation ™    1/34
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Part 24: Bayesian Estimation Bayesian Estimators p “Random Parameters” vs. Randomly Distributed Parameters p Models of Individual Heterogeneity n Random Effects: Consumer Brand Choice n Fixed Effects: Hospital Costs ™    2/34
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Part 24: Bayesian Estimation Bayesian Estimation p Specification of conditional likelihood: f(data | parameters) p Specification of priors: g(parameters) p Posterior density of parameters: p Posterior mean = E[parameters|data] ™    3/34 (data | parameters) (parameters) (parameters|data) (data) f g f f =
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Part 24: Bayesian Estimation The Marginal Density for the Data is Irrelevant ™    4/34 f(data| )p( ) L(data| )p( ) f( | data) f(data) f(data) Joint density of   and data is f(data, ) = L(data| )p( ) Marginal density of the data is           f(data)= f(data, )d L(data| )p( )d Thus,  f( | data β β β β β β β = = β β β β β β = β β β β L(data| )p( ) ) L(data| )p( )d  L(data| )p( )d Posterior Mean =   p( | data)d  L(data| )p( )d Requires specification of the likeihood and the prior. β β β β β β = β β β β β β β β β = β β β
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Part 24: Bayesian Estimation Computing Bayesian Estimators p First generation: Do the integration (math) p Contemporary - Simulation: n (1) Deduce the posterior n (2) Draw random samples of draws from the posterior and compute the sample means and variances of the samples. (Relies on the law of large numbers.) ™    5/34 (data | ) ( ) ( | data) (data) f g E d f β β β = β β
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Part 24: Bayesian Estimation Modeling Issues p As N , the likelihood dominates and the prior disappears  Bayesian and Classical MLE converge. (Needs the mode of the posterior to converge to the mean.) p Priors n Diffuse  large variances imply little prior information. (NONINFORMATIVE) n INFORMATIVE priors – finite variances that appear in the posterior. “Taints” any final results. ™    6/34
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Part 24: Bayesian Estimation A Practical Problem ™    7/34 + + - σ - - - - - σ μ π σ Γ + σ × - - σ - 2 2 v 1 2 v 2 2 vs (1/ ) K / 2 2 1 1/ 2 2 2 1 1 Sampling from the joint posterior may be impossible. E.g., linear regression. [vs ] 1 f( , | , ) e [2 ] | ( ) | (v 2) exp( (1/ 2)( ) [ ( ) ] ( )) What is this??? T y X X X β b X X b β β σ 2 o do 'simulation based estimation' here, we need joint observations on ( , ). β
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Part 24: Bayesian Estimation A Solution to the Sampling Problem ™    8/34 σ σ 2 2 The joint posterior, p( , |data) is intractable.  But, For inference about  , a sample from the marginal      posterior, p( |data) would suffice.
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