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Asymptotic Bias in Simulated Maximum Likelihood Estimation of Discrete Choice Models

Published online by Cambridge University Press:  11 February 2009

Lung-Fei Lee
Lung-Fei Lee
Affiliation:
The University of Michigan and Hong Kong University of Science and Technology
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Abstract

In this article, we investigate a bias in an asymptotic expansion of the simulated maximum likelihood estimator introduced by Lerman and Manski (pp. 305–319 in C. Manski and D. McFadden (eds.), Structural Analysis of Discrete Data with Econometric Applications, Cambridge: MIT Press, 1981) for the estimation of discrete choice models. This bias occurs due to the nonlinearity of the derivatives of the log likelihood function and the statistically independent simulation errors of the choice probabilities across observations. This bias can be the dominating bias in an asymptotic expansion of the simulated maximum likelihood estimator when the number of simulated random variables per observation does not increase at least as fast as the sample size. The properly normalized simulated maximum likelihood estimator even has an asymptotic bias in its limiting distribution if the number of simulated random variables increases only as fast as the square root of the sample size. A bias-adjustment is introduced that can reduce the bias. Some Monte Carlo experiments have demonstrated the usefulness of the bias-adjustment procedure.

Type
Articles
Information
Econometric Theory , Volume 11 , Issue 3 , June 1995 , pp. 437 - 483
Copyright
Copyright © Cambridge University Press 1995

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References

REFERENCES

Amemiya, T. (1985) Advanced Econometrics. Cambridge: Harvard University Press.Google Scholar
Borsch-Supan, A. & Hajivassiliou, V.A. (1993) Smooth unbiased multivariate probability simulators for maximum likelihood estimation of limited dependent variable models. Journal of Econometrics 58, 347–368.CrossRefGoogle Scholar
Cox, D.R. & Hinkley, D.V. (1974) Theoretical Statistics. Princeton, New Jersey: Princeton University Press.CrossRefGoogle Scholar
Gourieroux, C. & Monfort, A. (1993) Simulation-based inference: A survey with special reference to panel data models. Journal of Econometrics 59, 5–33.CrossRefGoogle Scholar
Hajivassiliou, V.A. (1993) Simulation estimation methods for limited dependent variable models. In Maddala, G.S., Rao, C.R., & Vinod, H.D. (eds.), Handbook of Statistics 11: Econometrics, ch. 19, pp. 519–543. Amsterdam: North-Holland.Google Scholar
Hajivassiliou, V.A. & McFadden, D. (1987) The Debt Repayment Crises LDC's: Estimation by the Method of Simulated Moments. Working paper, Yale University.Google Scholar
Keane, M.P. (1993) Simulation estimation for panel data models with limited dependent variables. In Maddala, G.S., Rao, C.R., & Vinod, H.D. (eds.), Handbook of Statistics 11: Econometrics, ch. 20, pp. 545–571. Amsterdam: North-Holland.Google Scholar
Keane, M.P. (1994) A computationally practical simulation estimator for panel data. Econometrica 62, 95–116.CrossRefGoogle Scholar
Lee, A.J. (1990) U-Statistics: Theory and Practice. New York: Marcel Dekker.Google Scholar
Lee, L.F. (1992) On efficiency of methods of simulated moments and maximum simulated likelihood estimation of discrete response models. Econometric Theory 8, 518–552.CrossRefGoogle Scholar
Lehmann, E.L. (1966) Nonparametrics: Statistical Methods Based on Ranks. San Francisco: Holden Day.Google Scholar
Lerman, S. & Manski, C. (1981) On the use of simulated frequencies to approximate choice probabilities. In Manski, C. & McFadden, D. (eds.), Structural Analysis of Discrete Data with Econometric Applications, ch. 7, pp. 305–319. Cambridge: MIT Press.Google Scholar
McFadden, D. (1989) A method of simulated moments for estimation of discrete response models without numerical integration. Econometrica 57, 995–1026.CrossRefGoogle Scholar
Pakes, A. (1986) Patents as options: Some estimates of the value of holding European patent stocks. Econometrica 54, 755–785.CrossRefGoogle Scholar
Pakes, A. & Pollard, D. (1989) Simulation and the asymptotics of optimization estimators. Econometrica 57, 1027–1057.CrossRefGoogle Scholar
Serfling, R.J. (1980) Approximation Theorems of Mathematical Statistics. New York: Wiley.CrossRefGoogle Scholar
Stern, S. (1992) A method for smoothing simulated moments of discrete probabilities in multivariate probit models. Econometrica 60, 943–952.CrossRefGoogle Scholar