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Functional Form and Heterogeneity in Models for Count Data


  • Greene, William


This study presents several extensions of the most familiar models for count data, the Poisson and negative binomial models. We develop an encompassing model for two well-known variants of the negative binomial model (the NB1 and NB2 forms). We then analyze some alternative approaches to the standard log gamma model for introducing heterogeneity into the loglinear conditional means for these models. The lognormal model provides a versatile alternative specification that is more flexible (and more natural) than the log gamma form, and provides a platform for several "two part" extensions, including zero inflation, hurdle, and sample selection models. (We briefly present some alternative approaches to modeling heterogeneity.) We also resolve some features in Hausman, Hall and Griliches (1984, Economic models for count data with an application to the patents–R&D relationship, Econometrica 52 , 909–938) widely used panel data treatments for the Poisson and negative binomial models that appear to conflict with more familiar models of fixed and random effects. Finally, we consider a bivariate Poisson model that is also based on the lognormal heterogeneity model. Two recent applications have used this model. We suggest that the correlation estimated in their model frameworks is an ambiguous measure of the correlation of the variables of interest, and may substantially overstate it. We conclude with a detailed application of the proposed methods using the data employed in one of the two aforementioned bivariate Poisson studies.

Suggested Citation

  • Greene, William, 2007. "Functional Form and Heterogeneity in Models for Count Data," Foundations and Trends(R) in Econometrics, now publishers, vol. 1(2), pages 113-218, August.
  • Handle: RePEc:now:fnteco:0800000008
    DOI: 10.1561/0800000008

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    References listed on IDEAS

    1. Wang, Peiming & Cockburn, Iain M & Puterman, Martin L, 1998. "Analysis of Patent Data--A Mixed-Poisson-Regression-Model Approach," Journal of Business & Economic Statistics, American Statistical Association, vol. 16(1), pages 27-41, January.
    2. Yen, Steven & Adamowicz, Wiktor L., 1994. "Participation, Trip Frequency and Site Choice: A Multinomial-Poisson Hurdle Model of Recreation Demand," Staff General Research Papers Archive 764, Iowa State University, Department of Economics.
    3. Rainer Winkelmann, 2004. "Health care reform and the number of doctor visits-an econometric analysis," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 19(4), pages 455-472.
    4. Heckman, James, 2013. "Sample selection bias as a specification error," Applied Econometrics, Russian Presidential Academy of National Economy and Public Administration (RANEPA), vol. 31(3), pages 129-137.
    5. Paul Contoyannis & Andrew M. Jones & Nigel Rice, 2004. "The dynamics of health in the British Household Panel Survey," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 19(4), pages 473-503.
    6. Jung, Robert C & Winkelmann, Rainer, 1993. "Two Aspects of Labor Mobility: A Bivariate Poisson Regression Approach," Empirical Economics, Springer, vol. 18(3), pages 543-556.
    7. Winkelmann, Rainer & Zimmermann, Klaus F, 1995. "Recent Developments in Count Data Modelling: Theory and Application," Journal of Economic Surveys, Wiley Blackwell, vol. 9(1), pages 1-24, March.
    8. William H. Greene, 1997. "FIML Estimation of Sample Selection Models for Count Data," Working Papers 97-02, New York University, Leonard N. Stern School of Business, Department of Economics.
    9. Murat K. Munkin & Pravin K. Trivedi, 1999. "Simulated maximum likelihood estimation of multivariate mixed-Poisson regression models, with application," Econometrics Journal, Royal Economic Society, vol. 2(1), pages 29-48.
    10. Vuong, Quang H, 1989. "Likelihood Ratio Tests for Model Selection and Non-nested Hypotheses," Econometrica, Econometric Society, vol. 57(2), pages 307-333, March.
    11. Joseph Hilbe, 1994. "Negative binomial regression," Stata Technical Bulletin, StataCorp LP, vol. 3(18).
    12. A. Colin Cameron & Pravin K. Trivedi, 1986. "Econometric models based on count data. Comparisons and applications of some estimators and tests," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 1(1), pages 29-53, January.
    13. Cameron, A. Colin & Trivedi, Pravin K., 1990. "Regression-based tests for overdispersion in the Poisson model," Journal of Econometrics, Elsevier, vol. 46(3), pages 347-364, December.
    14. William H. Greene, 1994. "Accounting for Excess Zeros and Sample Selection in Poisson and Negative Binomial Regression Models," Working Papers 94-10, New York University, Leonard N. Stern School of Business, Department of Economics.
    15. Andreas Million & Regina T. Riphahn & Achim Wambach, 2003. "Incentive effects in the demand for health care: a bivariate panel count data estimation," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 18(4), pages 387-405.
    16. William H. Greene, 1992. "A Statistical Model for Credit Scoring," Working Papers 92-29, New York University, Leonard N. Stern School of Business, Department of Economics.
    17. Terza, Joseph V., 1998. "Estimating count data models with endogenous switching: Sample selection and endogenous treatment effects," Journal of Econometrics, Elsevier, vol. 84(1), pages 129-154, May.
    18. Krinsky, Itzhak & Robb, A Leslie, 1986. "On Approximating the Statistical Properties of Elasticities," The Review of Economics and Statistics, MIT Press, vol. 68(4), pages 715-719, November.
    19. Daniel B. Hall, 2000. "Zero-Inflated Poisson and Binomial Regression with Random Effects: A Case Study," Biometrics, The International Biometric Society, vol. 56(4), pages 1030-1039, December.
    20. Freedman, David A., 2006. "On The So-Called "Huber-Sandwich Estimator" and "Robust Standard Errors"," The American Statistician, American Statistical Association, vol. 60, pages 299-302, November.
    21. Mullahy, John, 1986. "Specification and testing of some modified count data models," Journal of Econometrics, Elsevier, vol. 33(3), pages 341-365, December.
    22. Marcus Asplund & Rickard Sandin, 1999. "The Number of Firms and Production Capacity in Relation to Market Size," Journal of Industrial Economics, Wiley Blackwell, vol. 47(1), pages 69-85, March.
    23. Terza, Joseph V., 1985. "A Tobit-type estimator for the censored Poisson regression model," Economics Letters, Elsevier, vol. 18(4), pages 361-365.
    24. Cameron,A. Colin & Trivedi,Pravin K., 2005. "Microeconometrics," Cambridge Books, Cambridge University Press, number 9780521848053.
    25. Peiming Wang & Iain Cockburn & Martin L. Puterman, "undated". "A Mixed Poisson Regression Model for Analysis of Patent Data," Computing in Economics and Finance 1996 _049, Society for Computational Economics.
    26. Winkelmann, Rainer & Zimmermann, Klaus F., 1991. "A new approach for modeling economic count data," Economics Letters, Elsevier, vol. 37(2), pages 139-143, October.
    27. William H. Greene & Mark N. Harris & Bruce Hollingworth & Pushkar Maitra, 2008. "A Bivariate Latent Class Correlated Generalized Ordered Probit Model with an Application to Modeling Observed Obesity Levels," Working Papers 08-18, New York University, Leonard N. Stern School of Business, Department of Economics.
    28. Aigner, Dennis & Lovell, C. A. Knox & Schmidt, Peter, 1977. "Formulation and estimation of stochastic frontier production function models," Journal of Econometrics, Elsevier, vol. 6(1), pages 21-37, July.
    29. Butler, J S & Moffitt, Robert, 1982. "A Computationally Efficient Quadrature Procedure for the One-Factor Multinomial Probit Model," Econometrica, Econometric Society, vol. 50(3), pages 761-764, May.
    30. Weiren Wang & Felix Famoye, 1997. "Modeling household fertility decisions with generalized Poisson regression," Journal of Population Economics, Springer;European Society for Population Economics, vol. 10(3), pages 273-283.
    31. Gary King, 1989. "A Seemingly Unrelated Poisson Regression Model," Sociological Methods & Research, , vol. 17(3), pages 235-255, February.
    32. Lancaster, Tony, 2000. "The incidental parameter problem since 1948," Journal of Econometrics, Elsevier, vol. 95(2), pages 391-413, April.
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