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PanelDataNotes-12

# PanelDataNotes-12 - 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 12. Random Parameters Linear Models
Agenda ‘True’ Random Parameter Variation Discrete – Latent Class Continuous Classical Bayesian

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Parameter Heterogeneity i,t it it i,t (1) Regression model      y ε (2) Conditional probability model      f(y | x , ) (3) Heterogeneity - how are parameters distributed across      individuals?     (a)  Discrete - the populatio = + i,t i i x β β 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 ε σ j β 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 , , ) ε π π Σ π = = = σ j β

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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 ε ε ε = = = σ σ = σ j i j j β X , β β 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 , , ) ε = = = π σ i j X β ( 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 , , ) ε ε ε = = = σ σ = π σ 1 J j β β β
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 0 2 4 6 8 10 -4 Kernel density estim ate for YLC Density

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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 = = = = - μ - μ = = - φ ÷ ÷ ÷ ÷ σ σ σ σ π - μ ÷ = = - Σ ÷ ÷ ÷ σ σ π - μ ÷ = π - Σ ÷ σ σ π 2 ÷ ÷
Mixture of Normals +---------------------------------------------+ | Latent Class / Panel LinearRg Model | | Dependent variable YLC | | Number of observations 1000 | | Log likelihood function -1960.443 | | Info. Criterion: AIC =

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