Abstract
This paper develops finite mixture models with fixed effects for two families of distributions for which the incidental parameter problem has a solution. Analytical results are provided for mixtures of Normals and mixtures of Poisson. We provide algorithms based on the expectations-maximization (EM) approach as well as computationally simpler equivalent estimators that can be used in the case of the mixtures of normals. We design and implement a Monte Carlo study that examines the finite sample performance of the proposed estimator and also compares it with other estimators such the Mundlak-Chamberlain conditionally correlated random effects estimator. The results of Monte Carlo experiments suggest that our proposed estimators of such models have excellent finite sample properties, even in the case of relatively small T and moderately sized N dimensions. The methods are applied to models of healthcare expenditures and counts of utilization using data from the Health and Retirement Study.
We are grateful for comments and suggestions for improvement of an earlier version received from Anirban Basu, David Drukker, Maarten Lindeboom, and two anonymous reviewers and the editor. We also received helpful suggestions from participating audiences at 2011 SETA Conference and the 17th International Panel Data Conference. We alone are responsible for any errors.
References
Aitkin, M., and D. B. Rubin. 1985. “Estimation and Hypothesis Testing in Finite Mixture Models.” Journal of the Royal Statistical Society. Series B (Methodological) 47(1): 67–7510.1111/j.2517-6161.1985.tb01331.xSearch in Google Scholar
Allison, P. 2009. Fixed Effects Regression Models. Los Angeles: Sage Publications.10.4135/9781412993869Search in Google Scholar
Andersen, E. B. 1970. “Asymptotic Properties of Conditional Maximum Likelihood Estimators.” Journal of the Royal Statistical Society B 32: 283–301.10.1111/j.2517-6161.1970.tb00842.xSearch in Google Scholar
Arellano, M., and J. Hahn. 2007. “Understanding Bias in Nonlinear Panel Models: Some Recent Developments.” In Advances in Economics and Econometrics: Theory and Applications, Volume III, edited by R. Blundell, W. K. Newey, and T. Persson, 381–409. Cambridge: Cambridge University Press.10.1017/CBO9780511607547.013Search in Google Scholar
Bago d’Uva. T. 2005. “Latent Class Models for Use of Primary Care: Evidence From a British Panel.” Health Economics 14: 873–892.10.1002/hec.1047Search in Google Scholar
Cameron, A.C., and P.K Trivedi. 2005. Microeconometrics: Methods and Applications. Cambridge University Press.10.1017/CBO9780511811241Search in Google Scholar
Card, D., C. Dobkin, and N. Maestas. 2008. “The Impact of Nearly Universal Insurance Coverage on Health Care Utilization and Health: Evidence from Medicare.” American Economic Review 98(5): 2242–2258.10.1257/aer.98.5.2242Search in Google Scholar
Chamberlain, G. 1984. “Panel Data.” In Handbook of Econometrics, Volume II, edited by Z. Griliches, and M. Intriligator, 1247–1318. Amsterdam: North-Holland.10.1016/S1573-4412(84)02014-6Search in Google Scholar
Chamberlain, G. 1992. “Comment: Sequential Moment Restrictions in Panel Data.” Journal of Business and Economic Statistics 10: 20–26.Search in Google Scholar
Conway, K., and P. Deb. 2005. “Is Prenatal Care Really Ineffective? Or, is the ’Devil’ in the Distribution?.” Journal of Health Economics 24: 489–513.10.1016/j.jhealeco.2004.09.012Search in Google Scholar
Deb, P., and P. K. Trivedi 1997. “Demand for Medical Care by the Elderly: a Finite Mixture Approach.” Journal of Applied Econometrics 12: 313–336.10.1002/(SICI)1099-1255(199705)12:3<313::AID-JAE440>3.0.CO;2-GSearch in Google Scholar
Deb, P. and P. K. Trivedi. 2002. The Structure of Demand for Health Care: Latent Class versus Two-part Models.” Journal of Health Economics 21: 601–625.10.1016/S0167-6296(02)00008-5Search in Google Scholar
Dempster A. P., N. M. Laird, and D. B. Rubin. 1977. “Maximum Likelihood from Incomplete Data via EM Algorithm.” Journal of the Royal Statistical Society, B 39(1): 1–38.Search in Google Scholar
Dhaene, G., and K. Jochmans. 2011. Profile-score Adjustments for Nonlinear Fixed-effect Models. Paper presented at the 17th International Panel Data Conference, Montreal.Search in Google Scholar
Frühwirth-Schnatter, S. 2006. Finite Mixture and Markov Switching Models. New York: Springer.Search in Google Scholar
Greene, W. 2004a. “The Behavior of Fixed Effects Estimator in Nonlinear Models.” Econometrics Journal 7: 98–119.10.1111/j.1368-423X.2004.00123.xSearch in Google Scholar
Greene, W. 2004b. “Estimating Econometric Models with Fixed Effects.” Econometric Reviews 23: 125–147.10.1081/ETC-120039606Search in Google Scholar
Heckman, J. J. 1981. “The Incidental Parameters Problem and the Problem of Initial Conditions in estimating a Discrete Time-discrete Data Stochastic Process.” In Structural Analysis of Discrete Data with Econometric Applications, edited by C.F. Manski and D. McFadden 179–195. Cambridge, MA: MIT Press.Search in Google Scholar
Heckman, J. J., and B. Singer. 1984. “A Method for Minimizing the Impact of Distributional Assumptions in Econometric Models for Duration Data.” Econometrica 52: 271–320.10.2307/1911491Search in Google Scholar
Johnson R. W., and S. Crystal. 2000. “Uninsured Status and Out-of-Pocket Costs at Midlife.” Health Services Research 35(Part 1): 911–031.Search in Google Scholar
Lancaster, T. 2000. “The Incidental Parameter Problem Since 1948.” Journal of Econometrics 95: 391–413.10.1016/S0304-4076(99)00044-5Search in Google Scholar
Lancaster, T. 2002. “Orthogonal Parameters and Panel Data.” Review of Economic Studies 69: 647–666.10.1111/1467-937X.t01-1-00025Search in Google Scholar
Lindsay, B. G. 1995. Mixture Models: Theory, Geometry and Applications, NSF-CBMS Regional Conference Series in Probability and Statistics, volume 5, IMS-ASA.10.1214/cbms/1462106013Search in Google Scholar
Lindsay, B. G., and K. Roeder. 1992. “Residual Diagnostics in the Mixture Model.” Journal of American Statistical Association 87: 785–795.10.1080/01621459.1992.10475280Search in Google Scholar
Louis, T. A. 1982. “Finding the Observed Information Matrix when Using the EM Algorithm.” Journal of the Royal Statistical Society, B 44(2): 226–233.Search in Google Scholar
McLachlan, G., and D. Peel. 2004. Finite Mixture Models. New York: John Wiley.Search in Google Scholar
Mundlak, Y. 1978. “On the Pooling of Time Series and Cross Section Data.” Econometrica 46: 69–85.10.2307/1913646Search in Google Scholar
Oakes, D. 1999. “Direct Calculation of the Information Matrix via the EM Algorithm.” Journal of the Royal Statistical Society, B 61(2): 479–482.10.1111/1467-9868.00188Search in Google Scholar
Severini, T. A. 2000. Likelihood Methods in Statistics. Oxford: Oxford University Press.Search in Google Scholar
Skrondal, A., and S. Rabe-Hesketh. 2004. Generalized Latent Variable Modeling. New York: Chapman & Hall.10.1201/9780203489437Search in Google Scholar
Smith, J. P. 1998. “Socioeconomic Status and Health.” American Economic Review 88: 192–196.Search in Google Scholar
Smith, J. P., and R. Kington. 1997. “Demographic and Economic Correlates of Health in Old Age.” Demography 43: 159–170.10.2307/2061665Search in Google Scholar
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