Abstract
Linear regression models with random coefficients express the idea that each individual sampled may have a different linear response function. Technically speaking, random coefficient regression encompasses a rich variety of submodels. These include deconvolution or affine-mixture models as well as certain classical linear regression models that have heteroscedastic errors, or errors-in-variables, or random effects. This paper studies minimum distance estimates for the coefficient distributions in a general, semiparametric, random coefficient regression model. The analysis yields goodness-of-fit tests for the semiparametric model, prediction regions for future responses, and confidence regions for the distribution of the random coefficients.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Amemiya, T. (1977). A note on a heteroscedastic model,J. Econometrics,6, 365–370.
Begun, J. M., Hall, W. J., Huang, W.-M. and Wellner, J. A. (1983). Information and asymptotic efficiency in parametric-nonparametric models,Ann. Statist.,11, 432–452.
Beran, R. (1986). Simulated power functions,Ann. Statist.,14, 171–173.
Beran, R. (1988). Balanced simultaneous confidence sets,J. Amer. Statist. Assoc.,83, 679–686.
Beran, R. (1991). Designing bootstrap prediction bands,Nonparametric Functional Estimation and Related Topics (ed. G. Roussas), Nato ASI Series, 577–586, Kluwer, Dordrecht.
Beran, R. and Hall, P. (1992). Estimating coefficient distributions in random coefficient regressions,Ann. Statist.,20, 1970–1984.
Beran, R. and Millar, P. W. (1991). Minimum distance estimation in random coefficient regression models (unpublished).
Chow, G. C. (1983). Random and changing coefficient models,Handbook of Econometrics 2 (eds. Z. Griliches and M. D. Intriligator), North-Holland, Amsterdam.
Fan, J. (1991). On the optimal rates for nonparametric deconvolution problems,Ann. Statist.,19, 1257–1272.
Goldfeld, S. M. and Quandt, R. E. (1972).Nonlinear Methods in Econometrics, North-Holland, Amsterdam.
Hildreth, C. and Houck, J. P. (1968). Some estimators for a linear model with random coefficients,J. Amer. Statist. Assoc.,63, 584–595.
Hsiao, C. (1986).Analysis of Panel Data, Econometric Society Monographs, Cambridge Univ. Press, Cambridge.
LeCam, L. M. (1969).Théorie Asymptotique de la Décision Statistique, Univ. de Montréal.
Newbold, P. (1988). Some recent developments in time series analysis—III,Internat. Statist. Rev.,56, 17–29.
Nicholls, D. F. and Pagan, A. R. (1985). Varying coefficient regression,Handbook of Statistics 5 (eds. E. J. Hannan, P. R. Krishnaiah and M. M. Rao), 413–449, North-Holland, Amsterdam.
Pollard, D. (1980). The minimum distance method of testing,Metrika,27, 43–70.
Raj, B. and Ullah, A. (1981).Econometrics, A Varying Coefficients Approach, Croom-Helm, London.
Scheffé, H. (1959),The Analysis of Variance, Wiley, New York.
Schucany, W. R. and Wang, S. (1991). One-step bootstrapping for smooth iterative procedures,J. Roy. Statist. Soc. Ser. B,53, 587–596.
Spiegelman, C. (1979). On estimating the slope of a straight line when variables are subject to error,Ann. Statist.,7, 201–206.
van Es, A. J. (1991). Uniform deconvolution: nonparametric maximum likelihood and inverse estimation,Nonparametric Functional Estimation and Related Topics (ed. G. Roussas), NATO ASI Series, 191–198, Kluwer, Dordrecht.
Author information
Authors and Affiliations
Additional information
This research was supported in part by NSF Grant DMS 9001710.
About this article
Cite this article
Beran, R. Semiparametric random coefficient regression models. Ann Inst Stat Math 45, 639–654 (1993). https://doi.org/10.1007/BF00774778
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00774778