Hostname: page-component-745bb68f8f-grxwn Total loading time: 0 Render date: 2025-01-13T13:16:42.043Z Has data issue: false hasContentIssue false
  • English
  • Français

Using the Censored Gamma Distribution for Modeling Fractional Response Variables with an Application to Loss Given Default

Published online by Cambridge University Press:  09 August 2013

Fabio Sigrist  and
Werner A. Stahel
Werner A. Stahel
Affiliation:
Seminar for Statistics, Department of Mathematics, ETH Zurich, Rämistrasse 110, CH-8092 Zurich, Switzerland, E-mail: stahel@stat.math.ethz.ch
Get access Rights & Permissions [Opens in a new window]

Abstract

Regression models for limited continuous dependent variables having a non-negligible probability of attaining exactly their limits are presented. The models differ in the number of parameters and in their flexibility. Fractional data being a special case of limited dependent data, the models also apply to variables that are a fraction or a proportion. It is shown how to fit these models and they are applied to a Loss Given Default dataset from insurance to which they provide a good fit.

Keywords

Fractional response variablescensored distributionsTobit modelslimited dependent variablesLoss Given Default
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 41 , Issue 2 , November 2011 , pp. 673 - 710
Copyright
Copyright © International Actuarial Association 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramowitz, M. and Stegun, I.A. (1964) Handbook of Mathematical Functions. Dover Publications, New York.Google Scholar
Aitchison, J. (1955) On the distribution of a positive random variable having a discrete probability mass at the origin. J. Amer. Statist. Assoc, 50: 901–908.Google Scholar
Amemiya, T. (1985) Advanced Econometrics. Harvard University Press, Cambridge, Massachusetts.Google Scholar
Arabmazar, A. and Schmidt, P. (1982) An investigation of the robustness of the Tobit estimator to non-normality. Econometrica, 50(4): 1055–1063.CrossRefGoogle Scholar
Azzalini, A. and Capitanio, A. (2003) Distributions generated by per turbation of symmetry with emphasis on a multivariate skew t-distribution. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 65(2): 367–389.Google Scholar
Bardossy, A. and Plate, A.J. (1992) Space-time model for daily rainfall using atmospheric circulation patterns. Water Resources Research, 28(5): 1247–1259.CrossRefGoogle Scholar
Breen, R. (1996) Regression Models: Censored, Sample Selected, or Truncated Data. Sage Publications, Thousand Oaks.Google Scholar
Chambers, J.M. and Hastie, T.J. (1992) Statistical Models in S. Wadsworth & Brooks/Cole.Google Scholar
Chen, S. and Khan, S. (2001) Semiparametric estimation of a partially linear censored regression model. Econometric Theory, 17(03): 567–590.CrossRefGoogle Scholar
Cook, D.O., Kieschnick, R. and McCullough, B.D. (2008) Regression analysis of proportions in finance with self selection. Journal of Empirical Finance, 15(5): 860–867.Google Scholar
Couturier, D.L. and Victoria-Feser, M.-P. (2010) Zero-inflated trun cated generalized pareto distribution for the analysis of radio audience data. The Annals of Applied Statistics, 4(4): 1824–1846.Google Scholar
Cragg, J.G. (1971) Some statistical models for limited dependent variables with application to the demand for durable goods. Econometrica, 39(5): 829–44.Google Scholar
Crepon, B. and Duguet, E. (1997) Research and development, competition and innovation pseudo-maximum likelihood and simulated maximum likeli hood methods applied to count data models with heterogeneity. Journal of Econometrics, 79(2): 355–378.CrossRefGoogle Scholar
Deaton, A. and Irish, M. (1984) Statistical models for zero expenditures in household budgets. Journal of Public Economics, 23(1–2): 59–80.CrossRefGoogle Scholar
Dempster, A.P., Laird, N.M. and Rubin, D.B. (1977) Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society, Series B, 39(1): 1–38.Google Scholar
Fahrmeir, L. and Tutz, G. (2001) Multivariate statistical modelling based on generalized linear models. Springer Series in Statistics. Springer-Verlag, New York.Google Scholar
Ferrari, S. and Cribari-Neto, F. (2004) Beta regression for modelling rates and proportions. Journal of Applied Statistics, 31(7): 799–815.CrossRefGoogle Scholar
Goldberger, A.S. (1964) Economic Theory. Wiley, New York.Google Scholar
Gourieroux, C., Monfort, A. and Trognon, A. (1984) Pseudo maximum likelihood methods: Theory. Econometrica, 52(3): 681–700.Google Scholar
Gurmu, S. (1997) Semi-parametric estimation of hurdle regression models with an application to medicaid utilization. Journal of Applied Econometrics, 12: 225–242.Google Scholar
Gurmu, S. and Trivedi, P.K. (1996) Excess zeros in count models for recre ational trips. Journal of Business & Economic Statistics, 14(4): 469–77.Google Scholar
Heckman, J.J. (1976) The common structure of statistical models of truncation, sample selection and limited dependent variables and a simple estimator for such models. Annals of Economic and Social Measurement, 5(4): 120–137.Google Scholar
Heilbron, D.C. (1994) Zero-altered and other regression models for count data with added zeros. Biometrical Journal, 36: 531–547.Google Scholar
Khan, S. and Powell, J.L. (2001) Two-step estimation of semiparametric censored regression models. Journal of Econometrics, 103(1–2): 73–110.Google Scholar
Koenker, R. (2005) Quantile Regression, volume 1. Cambridge University Press, Cambridge University Press, 40 West 20th Street, New York.Google Scholar
Lambert, D. (1992) Zero-inflated Poisson regression, with an application to defects in manufacturing. Technometrics, 34: 1–14.CrossRefGoogle Scholar
Long, J.S. (1997) Regression Models for Categorical and Limited Dependent Variables. Advances quantiative techniques in the social sciences; v. 7. SAGE Publications, Inc., Thousand Oaks, California 91320.Google Scholar
Maddala, G.S. (1983) Limited-dependent and qualitative variables in econometrics, volume 3 of Econometric Society Monographs in Quantitative Economics. Cambridge University Press, Cambridge.Google Scholar
Maddala, G.S. and Nelson, F.D. (1975) Specification errors in limited dependent variable models. NBER Working Papers 0096, National Bureau of Economic Research, Inc. Google Scholar
McCullagh, P. and Nelder, J.A. (1983) Generalized linear models. Monographs on Statistics and Applied Probability. Chapman & Hall, London.Google Scholar
Miaou, S.-P. (1994) The relationship between truck accidents and geometric de sign of road sections: Poisson versus negative binomial regressions. Accident Analysis & Prevention, 26: 471–482.Google Scholar
Mullahy, J. (1986) Specification and testing of some modified count data models. Journal of Econometrics, 33(3): 341–365.CrossRefGoogle Scholar
Paolino, P. (2001) Maximum likelihood estimation of models with Beta-distributed dependent variables. Political Analysis, 9(4): 325–346.Google Scholar
Papke, L.E. and Wooldridge, J.M. (1996) Econometric methods for fractional response variables with an application to 401 (k) plan participation rates. Journal of Applied Econometrics, 11(6): 619–632.Google Scholar
Papke, L.E. and Wooldridge, J.M. (2008) Panel data methods for fractional response variables with an application to test pass rates. Journal of Econo metrics, 145(1–2): 121–133.CrossRefGoogle Scholar
Piessens, R., deDoncker-Kapenga, E., Uberhuber, C. and Kahaner, D. (1983) Quadpack. A subroutine package for automatic integration. Springer Series in Computational Mathematics, Volume 1. Springer-Verlag, New-York, 1983.Google Scholar
Powell, J.L. (1984) Least absolute deviations estimation for the censored regression model. Journal of Econometrics, 25(3): 303–325.Google Scholar
Ramalho, E.A., Ramalho, J.J.S. and Murteira, J.M.R. (2011) Alternative estimating and testing empirical strategies for fractional regression models. Journal of Economic Surveys, 25(1): 19–68.Google Scholar
Ramalho, J.J.S. and Vidigal da Silva, J. (2009) A two-part fractional regression model for the financial leverage decisions of micro, small, medium and large firms. Quantitative Finance, 9(5): 621–636.Google Scholar
Ridout, M., Demetrio, C.G.B. and Hinde, J. (1998) Models for count data with many zeros. In Proceedings of the XlXth International Biometrics Conference, pages 179–190, Cape Town, December 1998.Google Scholar
Rosett, R.N. and Nelson, F.D. (1975) Estimation of the two-limit probit regression model. Econometrica, 43(1): 141–46.Google Scholar
Sanso, B. and Guenni, L. (2004) A Bayesian approach to compare observed rainfall data to deterministic simulations. Environmetrics, 15(6): 597–612.Google Scholar
Shonkwiler, J.S. and Shaw, W.D. (1996) Hurdle count-data models in recreation demand analysis. Journal of Agricultural and Resource Economics, 21(02): 210–219.Google Scholar
Tobin, H. (1958) Estimation of relationships for limited dependent variables. Econometrica, 26: 24–36.Google Scholar
Vuong, Q.H. (1989) Likelihood ratio tests for model selection and non-nested hypotheses. Econometrica, 57(2): 307–333.Google Scholar
Welsh, A.H., Cunningham, R.B., Donnelly, C.F. and Lindenmayer, D.B. (1996) Modelling the abundance of rare species: statistical models for counts with extra zeros. Ecological Modelling, 88(1–3): 297–308.Google Scholar
Wooldridge, J.M. (2002) Econometric Analysis of Cross Section and Panel Data, 1st edition. MIT Press, Cambridge.Google Scholar
Wooldridge, J.M. (2010) Econometric Analysis of Cross Section and Panel Data, 2nd edition. MIT Press, Cambridge.Google Scholar