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A SIMPLE NONPARAMETRIC APPROACH FOR ESTIMATION AND INFERENCE OF CONDITIONAL QUANTILE FUNCTIONS

Published online by Cambridge University Press:  13 December 2021

Zheng Fang ,
Qi Li  and
Karen X. Yan
Zheng Fang
Affiliation:
Texas A&M University
Qi Li*
Affiliation:
Texas A&M University
Karen X. Yan
Affiliation:
Georgia Institute of Technology
*
Address correspondence to Qi Li, Department of Economics, Texas A&M University, College Station, TX, USA; e-mail [email protected].
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Abstract

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In this paper, we present a new nonparametric method for estimating a conditional quantile function and develop its weak convergence theory. The proposed estimator is computationally easy to implement and automatically ensures quantile monotonicity by construction. For inference, we propose to use a residual bootstrap method. Our Monte Carlo simulations show that this new estimator compares well with the check-function-based estimator in terms of estimation mean squared error. The bootstrap confidence bands yield adequate coverage probabilities. An empirical example uses a dataset of Canadian high school graduate earnings, illustrating the usefulness of the proposed method in applications.

Type
ARTICLES
Information
Econometric Theory , Volume 39 , Issue 2 , April 2023 , pp. 290 - 320
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Footnotes

We would like to thank the Editor, Peter Phillips, a Co-Editor, and three anonymous referees for helpful comments and suggestions, which have substantially improved the paper.

References

REFERENCES

Akritas, M.G. & Van Keilegom, I. (2001) Non-parametric estimation of the residual distribution. Scandinavian Journal of Statistics 28, 549–567.CrossRefGoogle Scholar
Angrist, J.D., Chernozhukov, V., & Fernández-Val, I. (2006) Quantile regression under misspecification, with an application to the U.S. wage structure. Econometrica 74, 539–563.CrossRefGoogle Scholar
Beran, R. (1982) Estimated sampling distributions: The bootstrap and competitors. Annals of Statistics 10, 212–225.CrossRefGoogle Scholar
Birke, M., Neumeyer, N., & Volgushev, S. (2017) The independence process in conditional quantile location-scale models and an application to testing for monotonicity. Statistica Sinica 27, 1815–1839.Google Scholar
Calonico, S., Cattaneo, M.D., & Farrell, M.H. (2018) On the effect of bias estimation on coverage accuracy in nonparametric inference. Journal of the American Statistical Association 113, 767–779.CrossRefGoogle Scholar
Calonico, S., Cattaneo, M.D., & Titiunik, R. (2014) Robust nonparametric confidence intervals for regression-discontinuity designs. Econometrica 82, 2295–2326.CrossRefGoogle Scholar
Chaudhuri, P. (1991) Nonparametric estimates of regression quantiles and their local bahadur representation. Annals of Statistics 19, 760–777.CrossRefGoogle Scholar
Chen, S., Dahl, G.B., & Khan, S. (2005) Nonparametric identification and estimation of a censored location-scale regression model. Journal of the American Statistical Association 100, 212–221.CrossRefGoogle Scholar
Chen, S. & Khan, S. (2008) Semiparametric estimation of nonstationary censored panel data models with time varying factor loads. Econometric Theory 24, 1149–1173.CrossRefGoogle Scholar
Cheng, F. (2002) Consistency of error density and distribution function estimators in nonparametric regression. Statistics and Probability Letters 59, 257–270.CrossRefGoogle Scholar
Cheng, G. & Huang, J. (2010) Bootstrap consistency for general semiparametric m-estimation. Annals of Statistics 38, 2884–2915.CrossRefGoogle Scholar
Chernozhukov, V., Fernández-Val, I., & Galichon, A. (2010) Quantile and probability curves without crossing. Econometrica 78, 1093–1125.Google Scholar
Dette, H. & Volgushev, S. (2008) Non-crossing non-parametric estimates of quantile curves. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 70, 609–627.CrossRefGoogle Scholar
Einmahl, J.H.J. & Van Keilegom, I. (2008) Specification tests in nonparametric regression. Journal of Econometrics 143, 88–102.CrossRefGoogle Scholar
Fan, J. & Gijbels, I. (1996) Local Polynomial Modelling and Its Applications. CRC Press.Google Scholar
Fan, J. & Yao, Q. (1998) Efficient estimation of conditional variance functions in stochastic regression. Biometrika 85, 645–660.CrossRefGoogle Scholar
Fang, Z. & Santos, A. (2018) Inference on directionally differentiable functions. The Review of Economic Studies 86, 377–412.Google Scholar
Florens, J.-P., Simar, L., & Keilegom, I.V. (2014) Frontier estimation in nonparametric location-scale models. Journal of Econometrics 178(Part 3), 456–470.Google Scholar
Hall, P. (1992) The Bootstrap and Edgeworth Expansion. Springer.CrossRefGoogle Scholar
Han, H., Linton, O., Oka, T., & Whang, Y.-J. (2016) The cross-quantilogram: Measuring quantile dependence and testing directional predictability between time series. Journal of Econometrics 193, 251–270.CrossRefGoogle Scholar
He, X. (1997) Quantile curves without crossing. The American Statistician 51, 186–192.Google Scholar
Heuchenne, C. & Keilegom, I.V. (2007) Location estimation in nonparametric regression with censored data. Journal of Multivariate Analysis 98, 1558–1582.CrossRefGoogle Scholar
Honda, T. (2004) Quantile regression in varying coefficient models. Journal of Statistical Planning and Inference 121, 113–125.CrossRefGoogle Scholar
Horowitz, J.L. & Lee, S. (2005) Nonparametric estimation of an additive quantile regression model. Journal of the American Statistical Association 100, 1238–1249.CrossRefGoogle Scholar
Horowitz, J.L. & Lee, S. (2007) Nonparametric instrumental variables estimation of a quantile regression model. Econometrica 75, 1191–1208.CrossRefGoogle Scholar
Kaplan, D.M. & Sun, Y. (2017) Smoothed estimating equations for instrumental variables quantile regression search within citing articles. Technical Report 1.CrossRefGoogle Scholar
Kiwitt, S. & Neumeyer, N. (2012) Estimating the conditional error distribution in non-parametric regression. Scandinavian Journal of Statistics 39, 259–281.CrossRefGoogle Scholar
Koenker, R., Ng, P., & Portnoy, S. (1994) Quantile smoothing splines. Biometrika 81, 673–680.CrossRefGoogle Scholar
Koenker, R. & Xiao, Z. (2002) Inference on the quantile regression process. Econometrica 70, 1583–1612.Google Scholar
Kosorok, M. (2008) Introduction to Empirical Processes and Semiparametric Inference. Springer.CrossRefGoogle Scholar
Li, D. & Li, Q. (2010) Nonparametric/semiparametric estimation and testing of econometric models with data dependent smoothing parameters. Journal of Econometrics 157, 179–190.CrossRefGoogle Scholar
Li, Q., Lin, J., & Racine, J.S. (2013) Optimal bandwidth selection for nonparametric conditional distribution and quantile functions. Journal of Business & Economic Statistics 31, 57–65.CrossRefGoogle Scholar
Mammen, E. (1991) Nonparametric regression under qualitative smoothness assumptions. Annals of Statistics 19, 741–759.CrossRefGoogle Scholar
Neumeyer, N. (2008) A bootstrap version of the residual-based smooth empirical distribution function. Journal of Nonparametric Statistics 20, 153–174.CrossRefGoogle Scholar
Neumeyer, N. (2009a) Smooth residual bootstrap for empirical processes of non-parametric regression residuals. Scandinavian Journal of Statistics 36, 204–228.CrossRefGoogle Scholar
Neumeyer, N. (2009b) Testing independence in nonparametric regression. Journal of Multivariate Analysis 100, 1551–1566.CrossRefGoogle Scholar
Phillips, P.C.B. (2015) Halbert White Jr. memorial JFEC lecture: Pitfalls and possibilities in predictive regression. Journal of Financial Econometrics 13, 521–555.CrossRefGoogle Scholar
Qu, Z. & Yoon, J. (2015) Nonparametric estimation and inference on conditional quantile processes. Journal of Econometrics 185, 1–19.CrossRefGoogle Scholar
Racine, J.S. & Li, K. (2017) Nonparametric conditional quantile estimation: A locally weighted quantile kernel approach. Journal of Econometrics 201, 72–94.CrossRefGoogle Scholar
Su, L. & Hoshino, T. (2016) Sieve instrumental variable quantile regression estimation of functional coefficient models. Journal of Econometrics 191, 231–254.CrossRefGoogle Scholar
van der Vaart, A.W. & Wellner, J.A. (1996) Weak Convergence and Empirical Processes. Springer.CrossRefGoogle Scholar
Van Keilegom, I. & Akritas, M.G. (1999) Transfer of tail information in censored regression. Annals of Statistics 27, 1745–1784.CrossRefGoogle Scholar
Yu, K. & Jones, M. (1998) Local linear quantile regression. Journal of the American Statistical Association 93, 228–237.CrossRefGoogle Scholar
Zhao, Z. & Xiao, Z. (2014) Efficient regressions via optimally combining quantile information. Econometric Theory 30, 1272–1314.CrossRefGoogle ScholarPubMed
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