Hostname: page-component-8448b6f56d-gtxcr Total loading time: 0 Render date: 2024-04-20T00:26:32.068Z Has data issue: false hasContentIssue false
  • English
  • Français

Variance Components Structures for the Extreme-Value and Logistic Distributions with Application to Models of Heterogeneity

Published online by Cambridge University Press:  11 February 2009

N. Scott Cardell
N. Scott Cardell
Affiliation:
Salford Systems
Get access Rights & Permissions [Opens in a new window]

Abstract

Two new classes of probability distributions are introduced that radically simplify the process of developing variance components structures for extremevalue and logistic distributions. When one of these new variates is added to an extreme-value (logistic) variate, the resulting distribution is also extreme value (logistic). Thus, quite complicated variance structures can be generated by recursively adding components having this new distribution, and the result will retain a marginal extreme-value (logistic) distribution. It is demonstrated that the computational simplicity of extreme-value error structures extends to the introduction of heterogeneity in duration, selection bias, limited-dependent- and qualitative-variable models. The usefulness of these new classes of distributions is illustrated with the examples of nested logit, multivariate risk, and competing risk models, where important generalizations to conventional stochastic structures are developed. The new models are shown to be computationally simpler and far more tractable than alternatives such as estimation by simulated moments. These results will be of considerable use to applied microeconomic researchers who have been hampered by computational difficulties in constructing more sophisticated estimators.

Type
Articles
Information
Econometric Theory , Volume 13 , Issue 2 , April 1997 , pp. 185 - 213
Copyright
Copyright © Cambridge University Press 1997

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

REFERENCES

Ahuja, J.C. & Nash, S.W. (1967) The generalized Gompertz-Verhulst family of distributions. Sankhya: The Indian Journal of Statistics: Series A 29, 141–156.Google Scholar
Beggs, S.D., Cardell, N.S., & Hausman, J. (1981) Assessing the potential demand for electric cars. Journal of Econometrics 17, 1–19.Google Scholar
Cardell, N.S. (1975) A Statistical Model Appropriate for a Statistical Test of the Independence of Irrelevant Alternatives Property of Multinomial Logit. Working paper, Charles River Associates, Cambridge, Massachusetts.Google Scholar
Cardell, N.S. (1989) Extensions of Multinomial Logit: The Hedonic Demand Model, the Nonindependent Logit Model, and the Ranked Logit Model. Unpublished Ph.D. Dissertation, Harvard University, Cambridge, Massachusetts.Google Scholar
Cardell, N.S. & Dunbar, F.C. (1980) Measuring the societal impacts of automobile downsizing. Transportation Research 14A, 389–404.Google Scholar
Cardell, N.S., Huang, S., & Brown, S. (1995) Decisionmaking in the motor carrier industry. Transportation Research 29A, 401–419.Google Scholar
Cardell, N.S. & Steinberg, D. (1992) A Gauss-Newton Method for Maximum Likelihood Estimation, with an Application to Nested Logit. Working paper, San Oiego State University, San Oiego.Google Scholar
Cramer, H. (1937) Random Variables and Probabilities Distributions. Cambridge Tracts in Mathematics and Mathematical Physics 36. Cambridge: Cambridge University Press.Google Scholar
Erdelyi, A. (ed.) (1953) Higher Transcendental Functions. Bateman Manuscript Project, vol. 1. New York: McGraw-Hill.Google Scholar
Feller, W. (1971) Introduction to Probability Theory and Its Applications, vol. 2, 2nd ed.New York: Wiley & Sons.Google Scholar
Gradshteyn, I.S. & Ryzhik, I.M. (1965) Table of Integrals, Series, and Products, 4th ed.New York: Academic Press.Google Scholar
Gumbel, E.J. (1961) Bivariate logistic distributions. American Statistical Association Journal 56, 335–349.Google Scholar
Han, A. & Hausman, J.A. (1990) Flexible parametric estimation of duration and competing risk models. Journal of Applied Econometrics 5, 1–28.Google Scholar
Harvard University (1994) Special section on simulation methods. Review of Economics and Statistics 76, 591–702.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 4, 475–492.Google Scholar
Heckman, J.J. (1978) Dummy endogenous variables in a simultaneous equation system. Econometrica 46, 931–959.CrossRefGoogle Scholar
Heckman, J.J. & Singer, B. (1986) Econometric analysis of longitudinal data. In Griliches, Z. & Intriligator, M.D. (eds.), Handbook of Econometrics, vol. 3, pp. 1689–1763. New York: North-Holland.Google Scholar
Hougaard, P. (1986) A class of multivariate failure time distributions. Biometrika 73, 671–678.Google Scholar
Huang, S. & Cardell, N.S. (1996) Duration Analysis on Multiple Welfare Spells: A Repeated Hazard Model with Heterogeneity. Working paper, Washington State University, Pullman.Google Scholar
Johnson, N.L. & Kotz, S. (1972) Distributions in Statistics: Continuous and Multivariate Distributions. New York: Wiley & Sons.Google Scholar
Kagan, A.M., Linnik, Y.V., & Rao, C.R. (1973) Characterization Problems in Mathematical Statistics. New York: Wiley & Sons.Google Scholar
Lukacs, E. (1970) Characteristic Functions, 2nd ed. London: Griffin.Google Scholar
McFadden, D. (1974) Conditional logit analysis of qualitative choice behavior. In Zarembka, P. (ed.), Frontiers in Econometrics, pp. 105–142. New York: Academic Press.Google Scholar
McFadden, D. (1981) Econometric models of probabilistic choice. In Manski, C.F. & McFadden, D. (eds.), Structural Analysis of Discrete Data with Econometric Applications, pp. 198–272. Cambridge, Massachusetts: MIT Press.Google Scholar
McFadden, D. (1984) Econometric analysis of qualitative response models. In Griliches, Z. & Intriligator, M.D. (eds.), Handbook of Econometrics, vol. 2, pp. 1395–1457. New York: North Holland.Google Scholar
Mundlak, Y. (1978) On the pooling of time series and cross section data. Econometrica 46, 69–85.Google Scholar
Oakes, D. & Manatunga, A.K. (1992) Fisher information for a bivariate extreme value distribution. Biometrika 79, 827–832.Google Scholar
Swamy, P.A.V.B. (1974) Linear models with random coefficients. In Zarembka, P. (ed.), Frontiers in Econometrics, pp. 143–168. New York: Academic Press.Google Scholar
Tawn, J.A. (1988) Bivariate extreme value theory: Models and estimation. Biometrika 75, 397–413.Google Scholar
Tawn, J.A. (1990) Modelling multivariate extreme value distributions. Biometrika 77, 245–253.Google Scholar
Tversky, A. (1972) Elimination-by-aspects: A theory of choice. Psychological Review 79, 281–299.Google Scholar
Verhulst, P.F. (1845) Recherches mathématiques sur la loi d'-accroissement de la population. Nouvelles Memoires de l'Academie Royale des Sciences et Belles-Lettres de Bruxelles 18, 38 pp.Google Scholar