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Published online by Cambridge University Press:  01 October 2009

S.C. Goh*
University of Toronto
K. Knight
University of Toronto
*Address correspondence to S.C. Goh, Department of Economics, University of Toronto, Max Gluskin House, 150 St. George Street, Toronto, Ontario, Canada M5S 3G7; e-mail:


It is well known that conventional Wald-type inference in the context of quantile regression is complicated by the need to construct estimates of the conditional densities of the response variables at the quantile of interest. This note explores the possibility of circumventing the need to construct conditional density estimates in this context with scale statistics that are explicitly inconsistent for the underlying conditional densities. This method of studentization leads conventional test statistics to have limiting distributions that are nonstandard but have the convenient feature of depending explicitly on the user’s choice of smoothing parameter. These limiting distributions depend on the distribution of the conditioning variables but can be straightforwardly approximated by resampling.

Copyright © Cambridge University Press 2009

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Bassett, G. & Koenker, R. (1978) Asymptotic theory of least absolute error regression. Journal of the American Statistical Association 73, 618622.CrossRefGoogle Scholar
Buchinsky, M. (1995) Estimating the asymptotic covariance matrix for quantile regression models: A Monte Carlo study. Journal of Econometrics 68, 303338.CrossRefGoogle Scholar
De Angelis, D., Hall, P., & Young, G.A. (1993) Analytical and bootstrap approximations to estimator distributions in L 1 regressions. Journal of the American Statistical Association 88, 13101316.Google Scholar
Gutenbrunner, C. & Jurečková, J. (1992) Regression rank scores and regression quantiles. Annals of Statistics 20, 305330.CrossRefGoogle Scholar
Hall, P. & Sheather, S.J. (1988) On the distribution of a Studentized quantile. Journal of the Royal Statistical Society, Series B 50, 381391.Google Scholar
Hendricks, W. & Koenker, R. (1992) Hierarchical spline models for conditional quantiles and the demand for electricity. Journal of the American Statistical Association 87, 5868.CrossRefGoogle Scholar
Horowitz, J.L. (1998) Bootstrap methods for median regression models. Econometrica 66, 13271351.CrossRefGoogle Scholar
Knight, K. & Goh, C. (2008) Asymptotics of Quantile Regression Basic Solutions with Applications. Mimeo, Department of Statistics, University of Toronto. An earlier version of this paper is available at Scholar
Koenker, R. (2005) Quantile Regression. Cambridge University Press.CrossRefGoogle Scholar
Koenker, R. & Machado, J.A.F. (1999) Goodness of fit and related inference processes for quantile regression. Journal of the American Statistical Association 94, 12961310.CrossRefGoogle Scholar
Koenker, R. & Xiao, Z. (2002) Inference on the quantile regression process. Econometrica 70, 15831612.CrossRefGoogle Scholar
Mukhin, A.B. (1985) Local limit theorems for distributions of sums of independent random vectors. Theory of Probability and Its Applications 29, 369375.CrossRefGoogle Scholar
Mukhin, A.B. (1991) Local limit theorems for lattice random variables. Theory of Probability and Its Applications 36, 698713.CrossRefGoogle Scholar
Portnoy, S. (1991) Asymptotic behavior of the number of regression quantile breakpoints. SIAM Journal on Scientific and Statistical Computing 12, 867883.CrossRefGoogle Scholar
Sakov, A. & Bickel, P.J. (2000) An Edgeworth expansion for the m out of n bootstrapped median. Statistics and Probability Letters 49, 217223.CrossRefGoogle Scholar