MLEfor HRCM
“Maximum Likelihood Estimates for the Hildreth-Houck Random Coefficients Model,” Econometrics Journal.Vol 5. No.1, p237-262, 2002
ABSTRACT
We explore maximum likelihood (ML) estimation of the Hildreth–Houck random coefficients model. We show that the global ML estimator can be inconsistent. We develop an alternative LML (local ML) estimator and prove that it is consistent and asymptotically efficient for points in the interior of the parameters. Properties of the LML and comparisons with common method of moments (MM) estimates are done via Monte Carlo. Boundary parameters lead to nonstandard asymptotic distributions for the LML which are described. The LML is used to develop a modification of the LR test for random coefficients. Simulations suggest that the LR test is more powerful for distant alternatives than the Breusch–Pagan (BP) Lagrange multiplier test. A simple modification of the BP test also appears to be more powerful than the BP.
REFERENCES
Amemiya, T. (1977). A note on a heteroskedastic model. Journal of Econometrics 6, 365–70. See also: (1978) Corrigenda, Journal of Econometrics 8, 265.
Andrews, D. W. K. (1999). Estimation when a parameter is on a boundary. Econometrica 67, 1341–83.
Bickel, P. J. (1978). Using residuals robustly I: tests for heteroscedasticity, nonlinearity. Annals of Statistics 6, 266–91.
Breusch, T. and A. Pagan (1979). A simple test of heteroskedasticity and random coefficient variation. Econometrica 47, 1287–94.
Chung, K. L. (1974). A Course in Probability Theory. New York: Academic. Cragg, J. G. (1984). More efficient estimation in the presence of heteroscedasticity of unknown form. Econometrica 51, 751–63.
Davidson, R. and J. G. MacKinnon (1992). Regression-based methods for using control variates in Monte Carlo experiments. Journal of Econometrics 54, 203–22.
Dent, W. T. and C. Hildreth (1977). Maximum likelihood estimation in random coefficient models. Journal of the American Statistical Association 72, 69–72.
Geyer, C. J. (1994). On the asymptotics of constrained M-estimation. Annals of Statistics 22, 1993–2010.
Ghosh, J. K. (1994). Higher Order Asymptotics NSF-CBMS Regional Conference Series in Probability and Statistics, vol. 4, IMS and ASA.
Gund ¨ uz, Y. B. (1999). Estimators of random coefficient models, Ph.D. Thesis, Bilkent University, Eco- ¨ nomics Department, 06533 Ankara, Turkey.
Gund ¨ uz, Y. B. and A. Zaman (1999). ¨ Monte Carlo Comparisons of Estimators for Random Coefficient Models. Technical report. Ankara: Economics Department, Bilkent University. c Royal Economic Society 2002 254
Asad Zaman Hamilton, J. D. (1991). A quasi-Bayesian approach to estimating parameters for mixtures of normal distributions. Journal of Business and Economic Statistics 9, 27–39.
Hendry, D. F. (1984). Monte Carlo experimentation in econometrics. In Z. Griliches and M. D. Intriligator (eds), Handbook of Econometrics, vol. II, pp. 937–74. Amsterdam: Elsevier.
Hendry, D. F. (1995). Dynamic Econometrics. Oxford: Oxford University Press.
Hildreth, C. and J. P. Houck (1968). Some estimators for a linear model with random coefficients. Journal of the American Statistical Association 63, 584–95. Ibragimov, I. A. and R. Z.
Has’minskii (1981). Statistical Estimation: Asymptotic Theory. New York: Springer. Kakwani, N. C. (1967). The unbiasedness of Zellner’s seemingly unrelated regression equations estimator. Journal of the American Statistical Association 62, 141–2.
Kiefer, J. and J. Wolfowitz (1956). Consistency of maximum likelihood estimator in presence of infinitely many nuisance parameters. Annals of Mathematical Statistics 27, 884–906.
Kiefer, N. M. (1978). Discrete parameter estimation of a switching regression model. Econometrica 46, 427–34.
Langford, N. T. (1993). Random Coefficient Models. Oxford: Clarendon. Robinson, P. M. (1987). Asymptotically efficient estimation in the presence of heteroskedasticity of unknown form. Econometrica 55, 875–91. Schmidt, P. (1976). Econometrics. New York:
Dekker. Schoenberg, R. (1996). Constrained maximum likelihood. Manuscript downloadable from http://weber.u.washington.edu/ rons/cmlinf/cmlinf.html.
Schwallie, D. P. (1982). Unconstrained maximum likelihood estimators of contemporaneous covariances. Economics Letters 9, 359–64.
Self, S. G. and K. Y. Liang (1987). Asymptotic properties of maximum likelihood estimators and likelihoood ratio tests under nonstandard conditions. Journal of the American Statistical Association 82, 605–10.
Swamy, P. and G. S. Tavlas (1995). Random coefficient models: theory and applications. Journal of Economic Surveys 9, 165–96.
Swamy, P. A. V. B. (1971). Statistical Inference in Random Coefficient Regression Models. New York: Springer.
White, H. (1980). A heteroskedasticity consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica 48, 817–38.
Zaman, A. (1989). Consistency via type 2 inequalities: a generalization of Wu’s theorem. Econometric Theory 5, 272–86.
Zaman, A. (1991). Asymptotic suprema of averaged random functions. Annals of Statistics 19, 2145–59.
Zaman, A. (1996). Statistical Foundations for Econometric Techniques. New York: Academic.
Zaman, A. (1998). Random Coefficients and Heteroskedasticity. Technical report 98-04, Economics Department, Bilkent University, 06533 Ankara, Turkey.
Zaman, A. (2000). Inconsistency of the Breusch–Pagan test. The Journal of Economic and Social Research 2:1, 1–11.