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UNIFORM BIAS STUDY AND BAHADUR REPRESENTATION FOR LOCAL POLYNOMIAL ESTIMATORS OF THE CONDITIONAL QUANTILE FUNCTION

Published online by Cambridge University Press:  02 August 2011

Emmanuel Guerre  and
Camille Sabbah
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Abstract

This paper investigates the bias and the weak Bahadur representation of a local polynomial estimator of the conditional quantile function and its derivatives. The bias and Bahadur remainder term are studied uniformly with respect to the quantile level, the covariates, and the smoothing parameter. The order of the local polynomial estimator can be higher than the differentiability order of the conditional quantile function. Applications of the results deal with global optimal consistency rates of the local polynomial quantile estimator, performance of random bandwidths, and estimation of the conditional quantile density function. The latter allows us to obtain a simple estimator of the conditional quantile function of the private values in a first-price sealed bids auction under the independent private values paradigm and risk neutrality.

Type
Research Article
Information
Econometric Theory , Volume 28 , Issue 1 , February 2012 , pp. 87 - 129
Copyright
Copyright © Cambridge University Press 2011

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Footnotes

This paper was started and completed when both authors were at Laboratoire de Statistique Théorique et Appliquée, Université Pierre et Marie Curie, support from which is gratefully acknowledged. Financial support from the Department of Economics, Queen Mary University of London, is also gratefully acknowledged. The authors thank the participants of the Queen Mary Econometrics Reading Group, of the Berlin Quantile Regression Workshop, and of the LSE Econometrics and Statistics Workshop in addition to the associate editor and two anonymous referees whose careful reading, suggestions, and comments helped to improve the paper. All remaining errors are our responsibility.

References

REFERENCES

Chauduri, P. (1991) Nonparametric estimates of regression quantiles and their local Bahadur representation. Annals of Statistics 19, 760–777.Google Scholar
Chernozhukov, V. & Hansen, C. (2005) An IV model of quantile treatments effects. Econometrica 73, 245–261.CrossRefGoogle Scholar
Chesher, A. (2003) Identification in nonseparable models. Econometrica 71, 1405–1441.Google Scholar
Chow, Y.S. & Teicher, H. (2003) Probability Theory. Independence, Interchangeability Martingales, 3rd ed. Springer-Verlag.Google Scholar
Echenique, F. & Komunjer, I. (2009) Testing models with multiple equilibria by quantile methods. Econometrica 77, 1281–1297.Google Scholar
Einmahl, U. & Mason, D.M. (2005) Uniform in bandwidth consistency of kernel-type function estimators. Annals of Statistics 33, 1380–1403.CrossRefGoogle Scholar
Fan, J. (1992) Design-adaptive nonparametric regression. Journal of the American Statistical Association 87, 998–1004.Google Scholar
Fan, J. & Gijbels, I. (1996) Local Polynomial Modeling and Its Applications. Chapman and Hall/CRC.Google Scholar
Fan, J., Heckman, N.E., & Wand, M.P. (1995) Local polynomial kernel regression for generalized linear model and quasi-likelihood functions. Journal of the American Statistical Association 90, 141–151.CrossRefGoogle Scholar
Firpo, S., Fortin, N., & Lemieux, T. (2009) Unconditional quantile regression. Econometrica 77, 953–973.Google Scholar
Goldenshluger, A. & Lepski, O. (2008) Universal pointwise selection rule in multivariate function estimation. Bernoulli 14, 1150–1190.CrossRefGoogle Scholar
Goldenshluger, A. & Lepski, O. (2009) Structural adaptation via -norm oracle inequalities. Probability Theory and Related Fields 143, 41–71.CrossRefGoogle Scholar
Guerre, E., Perrigne, I., & Vuong, Q. (2000) Optimal nonparametric estimation of first price auctions. Econometrica 68, 525–574.CrossRefGoogle Scholar
Guerre, E., Perrigne, I., & Vuong, Q. (2009) Nonparametric identification of risk aversion in first-price auctions under exclusion restrictions. Econometrica 77, 1193–1227.Google Scholar
Haile, P.A., Hong, H., & Shum, M. (2003) Nonparametric Tests for Common Values in First-Price Sealed-Bid Auctions. Cowles Foundation Discussion paper.Google Scholar
Hjort, N. & Pollard, D. (1993) Asymptotics for Minimisers of Convex Processes. Manuscript, Yale. http://www.stat.yale.edu/Pollard/Papers/.Google Scholar
Holderlein, S. & Mammen, E. (2007) Identification of marginal effects in nonseparable models without monotonicity. Econometrica 75, 1513–1518.CrossRefGoogle Scholar
Holderlein, S. & Mammen, E. (2009) Identification and estimation of local average derivatives in non-separable models without monotonicity. Econometrics Journal 12, 1–25.CrossRefGoogle Scholar
Hong, S.Y. (2003) Bahadur representation and its applications for local polynomial estimates in nonparametric M-regression. Journal of Nonparametric Statistics 15, 237–251.CrossRefGoogle Scholar
Imbens, G.W. & Newey, W.K. (2009) Identification and estimation of triangular simultaneous equations models without additivity. Econometrica 77, 1481–1512.Google Scholar
Koenker, R. (2005) Quantile Regression. Cambridge University Press.CrossRefGoogle Scholar
Kong, E., Linton, O., & Xia, Y. (2010) Uniform Bahadur representation for local polynomial estimates of M-regression and its application to the additive model. Econometric Theory 26, 1529–1564.CrossRefGoogle Scholar
Lee, K.L. & Lee, E.R. (2008) Kernel methods for estimating derivatives of conditional quantiles. Journal of the Korean Statistical Society 37, 365–373.CrossRefGoogle Scholar
Li, Q. & Racine, J.S. (2008) Nonparametric estimation of conditional CDF and quantile functions with mixed categorical and continuous data. Journal of Business & Economic Statistics 26, 423–434.CrossRefGoogle Scholar
Loader, C. (1999) Local Regression and Likelihood. Springer-Verlag.Google Scholar
Marmer, V. & Shneyerov, A. (2012) Quantile-based nonparametric inference for first-price auctions. Journal of Econometrics, in press.CrossRefGoogle Scholar
Massart, P. (2007) Concentration Inequalities and Model Selection. Lecture Notes in Mathematics 1896. Ecole d’Eté de Probabilités de Saint Flour XXXIII-2003, Picard, Jean (ed.). Springer-Verlag.Google Scholar
Parzen, E. (1979) Nonparametric statistical data modeling. Journal of the American Statistical Association 74, 105–121.Google Scholar
Rothe, C. (2010) Nonparametric estimation of distributional policy effects. Journal of Econometrics 155, 56–70.Google Scholar
Stone, C.J. (1982) Optimal global rates of convergence for nonparametric regression. Annals of Statistics 10, 1040–1053.Google Scholar
Truong, Y.K. (1989) Asymptotic properties of kernel estimators based on local medians. Annals of Statistics 17, 606–617.Google Scholar
Tsybakov, A.B. (1986) Robust reconstruction of functions by the local-approximation method. Problemy Peredachi Informatsii 22, 69–84.Google Scholar
van de Geer, S. (1999) Empirical Processes in M-Estimation. Cambridge University Press.Google Scholar
van der Vaart, A.W. (1998) Asymptotic Statistics. Cambridge University Press.Google Scholar
White, H. (1994) Estimation, Inference and Specification Analysis. Econometric Society Monographs. Cambridge University Press.CrossRefGoogle Scholar
Xiang, X. (1995) Estimation of conditional quantile density function. Journal of Nonparametric Statistics 4, 309–316.CrossRefGoogle Scholar
Zeidler, E. (1985) Nonlinear Functional Analysis and Its Applications, Vol. 1: Fixed-Point Theorems. Springer-Verlag.Google Scholar