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Nonparametric Kernel Estimation for Semiparametric Models

Published online by Cambridge University Press:  11 February 2009

Donald W.K. Andrews
Affiliation:
Cowles Foundation for Research in Economics, Yale University

Abstract

This paper presents a number of consistency results for nonparametric kernel estimators of density and regression functions and their derivatives. These results are particularly useful in semiparametric estimation and testing problems that rely on preliminary nonparametric estimators, as in Andrews (1994, Econometrica 62, 43–72). The results allow for near-epoch dependent, nonidentically distributed random variables, data-dependent bandwidth sequences, preliminary estimation of parameters (e.g., nonparametric regression based on residuals), and nonparametric regression on index functions.

Type
Articles
Copyright
Copyright © Cambridge University Press 1995

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References

REFERENCES

Andrews, D.W.K. (1985) A nearly independent, but non-strong mixing, triangular array. Journal of Applied Probability 21, 930934.CrossRefGoogle Scholar
Andrews, D.W.K. (1989) Stochastic Equicontinuity for Semiparametric Models. Unpublished manuscript, Cowles Foundation, Yale University.Google Scholar
Andrews, D.W.K. (1991a) An empirical process central limit theorem for dependent nonidentically distributed random variables. Journal of Multivariate Analysis 38, 187203.CrossRefGoogle Scholar
Andrews, D.W.K. (1991b) Asymptotic optimality of generalized CL, cross-validation, and generalized cross-validation in regression with heteroskedastic errors. Journal of Econometrics 47, 359377.CrossRefGoogle Scholar
Andrews, D.W.K. (1991c) Asymptotics for Kernel-Based Non-Orthogonal Semiparametric Estimators. Unpublished manuscript, Cowles Foundation, Yale University.Google Scholar
Andrews, D.W.K. (1994a) Asymptotics for semiparametric econometric models via stochastic equicontinuity. Econometrica 62, 4372.CrossRefGoogle Scholar
Andrews, D.W.K. (1994b) Empirical process methods in econometrics. In Engle, R.F. & McFadden, D. (eds.), Handbook of Econometrics, vol. IV, pp. 22472294. New York: North Holland.Google Scholar
Bartlett, M.S. (1963) Statistical estimation of density functions. Sankhya A 25, 245254.Google Scholar
Bhattacharya, P.K. (1967) Estimation of a probability density function and its derivatives. Sankhya A 29, 373382.Google Scholar
Bierens, H. J. (1981) Robust Methods and Asymptotic Theory. Lecture Notes in Economics and Mathematical Systems no. 192. Berlin: Springer.Google Scholar
Bierens, H.J. (1983) Uniform consistency of kernel estimators of a regression function under generalized conditions. Journal of the American Statistical Association 77, 699707.CrossRefGoogle Scholar
Bierens, H.J. (1987) Kernel estimators of regression functions. In Bewley, T.F. (ed.), Advances in Econometrics: Fifth World Congress, vol. I, pp. 99144. New York: Cambridge University Press.CrossRefGoogle Scholar
Bierens, H.J. (1990) Model-free asymptotically best forecasting of stationary economic time series. Econometric Theory 6, 348383.CrossRefGoogle Scholar
Billingsley, P. (1968) Convergence of Probability Measures. New York: Wiley.Google Scholar
Gallant, A.R. (1987) Nonlinear Statistical Models. New York: Wiley.CrossRefGoogle Scholar
Gallant, A.R. & White, H. (1988) A Unified Theory of Estimation and Inference for Nonlinear Dynamic Models. New York: Basil Blackwell.Google Scholar
Györfi, L., Härdle, W., Sarda, P., & Vieu, P. (1989) Nonparametric Curve Estimation from Time Series. Springer Lecture Notes in Statistics no. 60. New York: Springer-Verlag.CrossRefGoogle Scholar
Hall, P. & Heyde, C.C. (1980) Martingale Limit Theory and Its Applications. New York: Academic Press.Google Scholar
Härdle, W., Hall, P., & Marron, J.S. (1988): How far are automatically chosen regression smoothing parameters from their optimum? Journal of the American Statistical Association 83, 8695.Google Scholar
Ibragimov, I.A. (1962) Some limit theorems for stationary processes. Theory of Probability and Its Applications 7, 349382.CrossRefGoogle Scholar
Ichimura, H. (1985) Semiparametric Least Squares Estimation of Single Index Models. Unpublished manuscript, Department of Economics, University of Minnesota.Google Scholar
Ichimura, H. & Lee, L.-F. (1990) Semiparametric estimation of multiple index models. In Barnett, W.A., Powell, J.L., & Tauchen, G. (eds.), Nonparametric and Semiparametric Methods in Econometrics and Statistics, New York: Cambridge University Press.Google Scholar
Klein, R.W. & Spady, R.H. (1993) An efficient semiparametric estimator for discrete choice models. Econometrica 61, 387421.CrossRefGoogle Scholar
Li, K.-C. (1987) Asymptotic optimality for Cp, CL, cross-validation and generalized crossvalidation: Discrete index set. Annals of Statistics 15, 958975.CrossRefGoogle Scholar
McLeish, D.L. (1975a) A maximal inequality and dependent strong laws. Annals of Probability 3, 826836.CrossRefGoogle Scholar
McLeish, D.L. (1975b) Invariance principles for dependent variables. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 32, 165178.CrossRefGoogle Scholar
McLeish, D.L. (1977) On the invariance principle for nonstationary mixingales. Annals of Probability 5, 616621.CrossRefGoogle Scholar
Nadaraya, E.A. (1964) On estimating regression. Theory of Probability and Its Applications 9, 141142.CrossRefGoogle Scholar
Newey, W.K. (1988) Two-Step Estimation of Sample Selection Models. Unpublished manuscript, Department of Economics, Princeton University.Google Scholar
Parzen, E. (1962) On estimation of a probability density function and mode. Annals of Mathematical Statistics 33, 10651076.CrossRefGoogle Scholar
Powell, J.L. (1987) Semiparametric Estimation of Bivariate Latent Variables Models. SSRI Working paper 8704, University of Wisconsin, Madison.Google Scholar
Robinson, P.M. (1987) Asymptotically efficient estimation in the presence of heteroskedasticity of unknown form. Econometrica 55, 875891.CrossRefGoogle Scholar
Robinson, P.M. (1988) Root-n-consistent semiparametric regression. Econometrica 56, 931954.CrossRefGoogle Scholar
Rosenblatt, M. (1956) Remarks on some non-parametric estimates of a density function. Annals of Mathematical Statistics 27, 832837.CrossRefGoogle Scholar
Stone, C.J. (1980) Optimal rates of convergence for nonparametric estimators. Annals of Statistics 8, 13481360.CrossRefGoogle Scholar
Stone, C.J. (1982) Optimal global rates of convergence for nonparametric regression. Annals of Statistics 10, 10401053.CrossRefGoogle Scholar
Watson, G.S. (1964) Smooth regression analysis. Sankhya A 26, 359372.Google Scholar
Whang, Y.-J. & Andrews, D.W.K. (1993) Tests of model specification for parametric and semiparametric models. Journal of Econometrics 57, 277318.CrossRefGoogle Scholar