Hostname: page-component-7c8c6479df-ph5wq Total loading time: 0 Render date: 2024-03-28T00:05:29.969Z Has data issue: false hasContentIssue false

NONPARAMETRIC REGRESSION IN THE PRESENCE OF MEASUREMENT ERROR

Published online by Cambridge University Press:  01 December 2004

Susanne M. Schennach
Affiliation:
University of Chicago

Abstract

We introduce a nonparametric regression estimator that is consistent in the presence of measurement error in the explanatory variable when one repeated observation of the mismeasured regressor is available. The approach taken relies on a useful property of the Fourier transform, namely, its ability to convert complicated integral equations into simple algebraic equations. The proposed estimator is shown to be asymptotically normal, and its rate of convergence in probability is derived as a function of the smoothness of the densities and conditional expectations involved. The resulting rates are often comparable to kernel deconvolution estimators, which provide consistent estimation under the much stronger assumption that the density of the measurement error is known. The finite-sample properties of the estimator are investigated through Monte Carlo experiments.This work was made possible in part through financial support from the National Science Foundation via grant SES-0214068. The author is grateful to the referees and the co-editor for their helpful comments.

Type
Research Article
Copyright
© 2004 Cambridge University Press

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

Andrews, D.W.K. (1991) Asymptotic normality of series estimators for nonparametric and semiparametric regression models. Econometrica 59, 307345.Google Scholar
Ashenfelter, O. & A.B. Krueger (1994) Estimates of the economic returns to schooling from a new sample of twins. American Economic Review 84, 11571173.Google Scholar
Borus, M.E. & G. Nestel (1973) Response bias in reports of father's education and socioeconomic status. Journal of the American Statistical Association 68, 816820.Google Scholar
Bowles, S. (1972) Schooling and inequality from generation to generation. Journal of Political Economy 80, S219S251.Google Scholar
Carroll, R. & P. Hall (1988) Optimal rates of convergence for deconvolving a density. Journal of the American Statistical Association 83, 11841186.Google Scholar
Carroll, R.J., D. Ruppert, & L.A. Stefanski (1995) Measurement Error in Nonlinear Models. Chapman and Hall.
Fan, J. (1991a) Asymptotic normality for deconvolution kernel density estimators. Sankhyma: The Indian Journal of Statistics 53, series A, pt. 1, 97110.Google Scholar
Fan, J. (1991b) On the optimal rates of convergence for nonparametric deconvolution problems. Annals of Statistics 19, 12571272.Google Scholar
Fan, J. & Y.K. Truong (1993) Nonparametric regression with errors in variables. Annals of Statistics 21, 19001925.Google Scholar
Freeman, R.B. (1984) Longitudinal analysis of the effects of trade unions. Journal of Labor Economics 2, 126.Google Scholar
Gel'fand, I.M. & G.E. Shilov (1964) Generalized Functions. Academic Press.
Härdle, W. & O. Linton (1994) Applied nonparametric methods. In R. Engle & D. McFadden (eds.), Handbook of Econometrics vol. IV, pp. 22952339. Elsevier Science.
Hausman, J., W. Newey, & J. Powell (1995) Nonlinear errors in variables: Estimation of some Engel curves. Journal of Econometrics 65, 205233.Google Scholar
Li, T. & Q. Vuong (1998) Nonparametric estimation of the measurement error model using multiple indicators. Journal of Multivariate Analysis 65, 139165.Google Scholar
Lighthill, M.J. (1962) Introduction to Fourier Analysis and Generalized Function. Cambridge University Press.
Liu, M. & R. Taylor (1989) A consistent nonparametric density estimator for the deconvolution problem. Canadian Journal of Statistics 17, 427438.Google Scholar
Morey, E.R. & D.M. Waldman (1998) Measurement error in recreation demand models: The joint estimation of participation, site choice, and site characteristics. Journal of Environmental Economics and Management 35, 262276.Google Scholar
Politis, D.N. & J.P. Romano (1999) Multivariate density estimation with general flat-top kernels of infinite order. Journal of Multivariate Analysis 68, 125.Google Scholar
Rao, P. (1992) Identifiability in Stochastic Models. Academic Press.
Schennach, S.M. (2004) Estimation of nonlinear models with measurement error. Econometrica 72, 3375.Google Scholar