PanelDataNotes-22

# PanelDataNotes-22 - Econometric Analysis of Panel Data...

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Econometric Analysis of Panel Data William Greene Department of Economics Stern School of Business

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Econometric Analysis of Panel Data 22. Individual Heterogeneity and Random Parameter Variation
Heterogeneity O bservational: Observable differences across  individuals (e.g., choice makers) C hoice strategy:  How consumers make  decisions – the underlying behavior S tructural: Differences in model frameworks P references: Differences in model ‘parameters’

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Parameter Heterogeneity i,t it it i,t (1) Regression model      y ε (2) Conditional probability or other nonlinear model      f(y | x , ) (3) Heterogeneity - how are parameters distributed across      individuals?     (a)  Discr = + i,t i i x β β ete - the population contains a mixture of Q           types of individuals.     (b)  Continuous. Parameters are part of the stochastic           structure of the population.
Distinguish Bayes and Classical Both depart from the heterogeneous ‘model,’ f(y it | x it )=g(y it , x it , β i ) What do we mean by ‘randomness’ With respect to the information of the analyst (Bayesian) With respect to some stochastic process governing ‘nature’  (Classical) Bayesian: No difference between ‘fixed’ and ‘random’ Classical: Full specification of joint distributions for  observed random variables; piecemeal definitions of   ‘random’ parameters.  Usually a form of ‘random effects’

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Hierarchical Bayesian Estimation i,t i,t 0 Sample data generation:  f(y | ) g(y , ) Individual heterogeneity:   ,   ~ N[ ] What information exists about 'the model?' p( ) =  N[ , ] = + i,t i,t i i i i 0 x , x , β Ω = u u 0, β β Γ Prior densities for structural parameters : β β Σ 0 ,  e.g.,   and (large) v p( ) =  Inverse Wishart[ , ] p( ) =  whatever works for other parameters in model p( )=  N[ , ] End Result: Joint prior distribution for all param γ i 0 I A Γ Ω Priors for parameters of interest : β β Γ 0 eters p( , , , |prior 'beliefs' in  , , , assumed densities) γ 0 i A,  β Γ Ω β β Σ
Allenby and Rossi: Structure it,j it,j it,j it,j Conditional data generation mechanism y * = + , Utility for consumer i, choice t, ε  brand j. (Consumer choice among brands of ketchup - the 'scanner data') Y   = 1[y * =maximum utility among i it, j x β it,j it,j j 1  the J choices] x = (constant, log price, "availability," "featured") ~ N[0, ], = 1 ε λ λ Implies a J outcome multinomial probit model.

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Priors i i i i β j j j β -1 β Prior Densities        ~N[ , ],         Implies  = + , ~ N[0, ]        ~ Inverse Gamma[v,s ] (looks like c λ hi-squared), v= 3, s = 1 Priors over structural model parameters        ~N[ ,a ],  = 0        β V β β w w V β β V β β β V 0 0 0 0 ~Wishart[v , ],v = 8, = 8 V V I
Bayesian Posterior Analysis Estimation of posterior distributions for upper level

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