Part 12: Random Parameters [ 1/46] Econometric Analysis of Panel Data William Greene Department of Economics Stern School of Business

Econometric Analysis of Panel Data 12. Random Parameters Linear Models

Part 12: Random Parameters [ 3/46] Parameter Heterogeneity

Part 12: Random Parameters [ 4/46] Agenda  ‘True’ Random Parameter Variation Discrete – Latent Class Continuous  Classical  Bayesian

Part 12: Random Parameters [ 5/46] Discrete Parameter Variation

Part 12: Random Parameters [ 6/46] Log Likelihood for an LC Model

Part 12: Random Parameters [ 7/46]

Part 12: Random Parameters [ 8/46] Example: Mixture of Normals

Part 12: Random Parameters [ 9/46] 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$

Part 12: Random Parameters [ 10/46] Mixture of Normals

Part 12: Random Parameters [ 11/46] Estimating Which Class

Part 12: Random Parameters [ 12/46] Posterior for Normal Mixture

Part 12: Random Parameters [ 13/46] Estimated Posterior Probabilities

Part 12: Random Parameters [ 14/46] More Difficult When the Populations are Close Together

Part 12: Random Parameters [ 15/46] The Technique Still Works Latent Class / Panel LinearRg Model Dependent variable YLC Sample is 1 pds and 1000 individuals LINEAR regression model Model fit with 2 latent classes Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X |Model parameters for latent class 1 Constant| *** Sigma| *** |Model parameters for latent class 2 Constant|.90156*** Sigma|.86951*** |Estimated prior probabilities for class membership Class1Pr|.73447*** Class2Pr|.26553***

Part 12: Random Parameters [ 16/46] Predicting Class Membership Means = 1 and 5 Means = 1 and |Cross Tabulation ||Cross Tabulation | | | | CLASS || | | CLASS | |CLASS1 | Total | 0 1 ||CLASS1 | Total | 0 1 | | 0| 787 | || 0| 787 | | | 1| 1713 | || 1| 1713 | | | Total| 2500 | || Total| 2500 | | Note: This is generally not possible as the true underlying class membership is not known.

Part 12: Random Parameters [ 17/46] How Many Classes?

Part 12: Random Parameters [ 18/46]

Part 12: Random Parameters [ 19/46] Latent Class Regression

Part 12: Random Parameters [ 20/46] An Extended Latent Class Model

Part 12: Random Parameters [ 21/46] Baltagi and Griffin’s Gasoline Data World Gasoline Demand Data, 18 OECD Countries, 19 years Variables in the file are COUNTRY = name of country YEAR = year, LGASPCAR = log of consumption per car LINCOMEP = log of per capita income LRPMG = log of real price of gasoline LCARPCAP = log of per capita number of cars See Baltagi (2001, p. 24) for analysis of these data. The article on which the analysis is based is Baltagi, B. and Griffin, J., "Gasoline Demand in the OECD: An Application of Pooling and Testing Procedures," European Economic Review, 22, 1983, pp The data were downloaded from the website for Baltagi's text.

Part 12: Random Parameters [ 22/46] 3 Class Linear Gasoline Model

Part 12: Random Parameters [ 23/46] Estimating E[β i |X i,y i, β 1 …, β Q ]

Part 12: Random Parameters [ 24/46] Estimated Parameters LCM vs. Gen1 RPM

Part 12: Random Parameters [ 25/46] Heckman and Singer’s RE Model  Random Effects Model  Random Constants with Discrete Distribution

Part 12: Random Parameters [ 26/46] LC Regression for Doctor Visits

Part 12: Random Parameters [ 27/46] 3 Class Heckman-Singer Form

Part 12: Random Parameters [ 28/46] The EM Algorithm

Part 12: Random Parameters [ 29/46] Implementing EM for LC Models

Part 12: Random Parameters [ 30/46] Continuous Parameter Variation (The Random Parameters Model)

Part 12: Random Parameters [ 31/46] OLS and GLS Are Consistent

Part 12: Random Parameters [ 32/46] ML Estimation of the RPM

Part 12: Random Parameters [ 33/46] RP Gasoline Market

Part 12: Random Parameters [ 34/46] Parameter Covariance matrix

Part 12: Random Parameters [ 35/46] RP vs. Gen1

Part 12: Random Parameters [ 36/46] Modeling Parameter Heterogeneity

Part 12: Random Parameters [ 37/46] Hierarchical Linear Model COUNTRY = name of country YEAR = year, LGASPCAR = log of consumption per cary LINCOMEP = log of per capita incomez LRPMG = log of real price of gasoline x1 LCARPCAP = log of per capita number of cars x2 y it =  1i +  2i x1 it +  3i x2 it +  it.  1i =  1 +  1 z i + u 1i  2i =  2 +  2 z i + u 2i  3i =  3 +  3 z i + u 3i

Part 12: Random Parameters [ 38/46] Estimated HLM

Part 12: Random Parameters [ 39/46] RP vs. HLM

Part 12: Random Parameters [ 40/46] A Hierarchical Linear Model German Health Care Data Hsat = β 1 + β 2 AGE it + γ i EDUC it + β 4 MARRIED it + ε it γ i = α 1 + α 2 FEMALE i + u i Sample ; all $ Setpanel ; Group = id ; Pds = ti $ Regress ; For [ti = 7] ; Lhs = newhsat ; Rhs = one,age,educ,married ; RPM = female ; Fcn = educ(n) ; pts = 25 ; halton ; panel ; Parameters$ Sample ; 1 – 887 $ Create ; betaeduc = beta_i $ Dstat ; rhs = betaeduc $ Histogram ; Rhs = betaeduc $

Part 12: Random Parameters [ 41/46] OLS Results OLS Starting values for random parameters model... Ordinary least squares regression LHS=NEWHSAT Mean = Standard deviation = Number of observs. = 6209 Model size Parameters = 4 Degrees of freedom = 6205 Residuals Sum of squares = Standard error of e = Fit R-squared = Adjusted R-squared = Model test F[ 3, 6205] (prob) = 142.0(.0000) | Standard Prob. Mean NEWHSAT| Coefficient Error z z>|Z| of X Constant| *** AGE| *** MARRIED|.29664*** EDUC|.14464***

Part 12: Random Parameters [ 42/46] Maximum Simulated Likelihood Random Coefficients LinearRg Model Dependent variable NEWHSAT Log likelihood function Estimation based on N = 6209, K = 7 Unbalanced panel has 887 individuals LINEAR regression model Simulation based on 25 Halton draws | Standard Prob. Mean NEWHSAT| Coefficient Error z z>|Z| of X |Nonrandom parameters Constant| *** AGE| *** MARRIED|.23427*** |Means for random parameters EDUC|.16580*** |Scale parameters for dists. of random parameters EDUC| *** |Heterogeneity in the means of random parameters cEDU_FEM| *** |Variance parameter given is sigma Std.Dev.| ***

Part 12: Random Parameters [ 43/46] Simulating Conditional Means for Individual Parameters Posterior estimates of E[parameters(i) | Data(i)]

Part 12: Random Parameters [ 44/46] “Individual Coefficients” --> Sample ; $ --> create ; betaeduc = beta_i $ --> dstat ; rhs = betaeduc $ Descriptive Statistics All results based on nonmissing observations. ============================================================================== Variable Mean Std.Dev. Minimum Maximum Cases Missing ============================================================================== All observations in current sample BETAEDUC|

Part 12: Random Parameters [ 45/46] Hierarchical Bayesian Estimation

Part 12: Random Parameters [ 46/46] Estimation of Hierarchical Bayes Models