Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-27T17:26:21.706Z Has data issue: false hasContentIssue false
  • English
  • Français

CONFIDENCE BANDS IN QUANTILE REGRESSION

Published online by Cambridge University Press:  04 November 2009

Wolfgang K. Härdle  and
Song Song
Wolfgang K. Härdle
Affiliation:
Humboldt-Universität zu Berlin
Song Song*
Affiliation:
Humboldt-Universität zu Berlin
*
*Address correspondence to Song Song, Institute for Statistics and Econometrics, Humboldt-Universität zu Berlin, Spandauer Straße 1, 10178 Berlin, Germany; e-mail: [email protected].
Get access Rights & Permissions [Opens in a new window]

Abstract

Let (X1, Y1), …, (Xn, Yn) be independent and identically distributed random variables and let l(x) be the unknown p-quantile regression curve of Y conditional on X. A quantile smoother ln(x) is a localized, nonlinear estimator of l(x). The strong uniform consistency rate is established under general conditions. In many applications it is necessary to know the stochastic fluctuation of the process {ln(x) – l(x)}. Using strong approximations of the empirical process and extreme value theory, we consider the asymptotic maximal deviation sup0≤x≤1 |ln(x) − l(x)|. The derived result helps in the construction of a uniform confidence band for the quantile curve l(x). This confidence band can be applied as a econometric model check. An economic application considers the relation between age and earnings in the labor market by means of parametric model specification tests, which presents a new framework to describe trends in the entire wage distribution in a parsimonious way.

Type
ARTICLES
Information
Econometric Theory , Volume 26 , Issue 4 , August 2010 , pp. 1180 - 1200
Copyright
Copyright © Cambridge University Press 2009

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

Bickel, P. & Rosenblatt, M. (1973) On some global measures of the deviation of density function estimatiors. Annals of Statistics 1, 1071–1095.CrossRefGoogle Scholar
Cai, Z.W. (2002) Regression quantiles for time series. Econometric Theory 18, 169–192.CrossRefGoogle Scholar
Csörgö, S. & Hall, P. (1982) Upper and lower classes for triangular arrays. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 61, 207–222.CrossRefGoogle Scholar
Fan, J., Hu, T.C., & Troung, Y.K. (1994) Robust nonparametric function estimation. Scandinavian Journal of Statistics 21, 433–446.Google Scholar
Fan, J., Yao, Q., & Tong, H. (1996) Estimation of conditional densities and sensitivity measures in nonlinear dynamical systems. Biometrika 83, 189–206.CrossRefGoogle Scholar
Ferguson, T.S. (1967) Mathematical Statistics: A Decision Theoretic Approach. Academic Press.Google Scholar
Franke, J. & Mwita, P. (2003) Nonparametric Estimates for Conditional Quantiles of Time Series. Report in Wirtschaftsmathematik 87, University of Kaiserslautern.Google Scholar
Hall, P., Wolff, R., & Yao, Q. (1999) Methods for estimating a conditional distribution function. Journal of the American Statistical Association 94, 154–163.CrossRefGoogle Scholar
Härdle, W. (1989) Asymptotic maximal deviation of M-smoothers. Journal of Multivariate Analysis 29, 163–179.CrossRefGoogle Scholar
Härdle, W., Janssen, P. & Serfling, R. (1988) Strong uniform consistency rates for estimators of conditional functionals. Annals of Statistics 16, 1428–1429.CrossRefGoogle Scholar
Härdle, W. & Luckhaus, S. (1984) Uniform consistency of a class of regression function estimators. Annals of Statistics 12, 612–623.CrossRefGoogle Scholar
Huber, P. (1981) Robust Statistics. Wiley.CrossRefGoogle Scholar
Jeong, K. & Härdle, W. (2008) A Consistent Nonparametric Test for Causality in Quantile. SFB 649 Discussion Paper.CrossRefGoogle Scholar
Johnston, G. (1982) Probabilities of maximal deviations of nonparametric regression function estimates. Journal of Multivariate Analysis 12, 402–414.CrossRefGoogle Scholar
Koenker, R. & Bassett, G.W. (1978) Regression quantiles. Econometrica 46, 33–50.CrossRefGoogle Scholar
Koenker, R. & Hallock, K.F. (2001) Quantile regression. Journal of Econometric Perspectives 15, 143–156.CrossRefGoogle Scholar
Koenker, R. & Park, B.J. (1996) An interior point algorithm for nonlinear quantile regression. Journal of Econometrics 71, 265–283.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, forthcoming.CrossRefGoogle Scholar
Lejeune, M.G. & Sarda, P. (1988) Quantile regression: A nonparametric approach. Computational Statistics and Data Analysis 6, 229–239.CrossRefGoogle Scholar
Murphy, K. & Welch, F. (1990) Empirical age-earnings profiles. Journal of Labor Economics 8, 202–229.CrossRefGoogle Scholar
Parzen, M. (1962) On estimation of a probability density function and mode. Annals of Mathematical Statistics 32, 1065–1076.CrossRefGoogle Scholar
Portnoy, S. & Koenker, R. (1997) The Gaussian hare and the Laplacian tortoise: Computability of squared-error versus absolute-error estimators (with discussion). Statistical Sciences 12, 279– 300.CrossRefGoogle Scholar
Rosenblatt, M. (1952) Remarks on a multivariate transformation. Annals of Mathematical Statistics 23, 470–472.CrossRefGoogle Scholar
Ruppert, D., Sheather, S.J., & Wand, M.P. (1995) An effective bandwidth selector for local least squares regression. Journal of the American Statistical Association 90, 1257–1270.CrossRefGoogle Scholar
Tusnady, G. (1977) A remark on the approximation of the sample distribution function in the multidimensional case. Periodica Mathematica Hungarica 8, 53–55.CrossRefGoogle Scholar
Yu, K. & Jones, M.C. (1997) A comparison of local constant and local linear regression quantile estimation. Computational Statistics and Data Analysis 25, 159–166.CrossRefGoogle Scholar
Yu, K. & Jones, M.C. (1998) Local linear quantile regression. Journal of the American Statistical Association 93, 228–237.CrossRefGoogle Scholar
Yu, K., Lu, Z., & Stander, J. (2003) Quantile regression: Applications and current research areas. Journal of the Royal Statistical Society, Series D 52, 331–350.Google Scholar