PanelDataNotes-12

PanelDataNotes-12 - Econometric Analysis of Panel Data...

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Unformatted text preview: Econometric Analysis of Panel Data William Greene Department of Economics Stern School of Business Econometric Analysis of Panel Data 12. Random Parameters Linear Models Agenda True Random Parameter Variation Discrete Latent Class Continuous Classical Bayesian Parameter Heterogeneity i,t it it i,t (1) Regression model y (2) Conditional probability model f(y | x , ) (3) Heterogeneity - how are parameters distributed across individuals? (a) Discrete - the populatio = + i,t i i x n contains a mixture of J types of individuals. (b) Continuous. Parameters are part of the stochastic structure of the population. Discrete Parameter Variation ,j The Latent Class Model (1) Population is a (finite) mixture of J types of individuals. j = 1,...,J. J 'classes' differentiated by ( , ) (a) Analyst does not know class memberships. ('latent j J 1 J j= 1 J i,t it i,t ,j .') (b) 'Mixing probabilities' (from the point of view of the analyst) are ,..., , with 1 (2) Conditional density is P(y | class j) f(y | x , , ) = = = j Estimating an LC Model i i i,t it i,t ,j i T i1 i2 i,T ,j it i,t ,j t 1 i Conditional density for each observation is P(y | class j) f(y | x , , ) Joint conditional density for T observations is f(y , y ,..., y | , ) f(y | x , , ) (T may be 1. Th = = = = j i j j X , i T J i1 i2 i,T j it i,t ,j j 1 t 1 is is not only a 'panel data' model.) Maximize this for each class if the classes are known. They aren't. Unconditional density for individual i is f(y , y ,..., y | ) f(y | x , , ) = = = i j X ( 29 ( 29 i i ,1 ,J T N J j it i,t ,j i 1 j 1 t 1 LogLikelihood LogL(( , ),...,( , )) log f(y | x , , ) = = = = 1 J j Unmixing a Mixed Sample Sample ; 1 1000$ Calc ; Ran(123457)$ Create ; lc1=rnn(1,1) ;lc2=rnn(5,1)$ Create ; class=rnu(0,1)$ Create ; if(class<.3)ylc=lc1 ; (else)ylc=lc2$ Kernel ; rhs=ylc $ Regress; lhs=ylc;rhs=one;lcm;pts=2;pds=1$ YLC .045 .090 .135 .180 .224 .000-2 2 4 6 8 10-4 Kernel density estim ate for YLC Density Mixture of Normals 2 it j it j it j j j j T 2 it j T i1 iT t 1 j j T N J it j T j t 1 i 1 j 1 j j y y 1 1 1 f(y | class j) exp = 2 2 y 1 1 f(y ,..., y | class j) exp 2 2 y 1 1 logL log exp 2 2 = = = = - - = =- - = =- - = - 2...
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PanelDataNotes-12 - Econometric Analysis of Panel Data...

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