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Smoothed Empirical Likelihood Methods for Quantile Regression Models

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  • Yoon-Jae Whang

    (Korea University)

Abstract

This paper considers an empirical likelihood method to estimate the parameters of the quantile regression (QR) models and to construct confidence regions that are accurate in finite samples. To achieve the higher-order refinements, we smooth the estimating equations for the empirical likelihood. We show that the smoothed empirical likelihood (SEL) estimator is first-order asymptotically equivalent to the standard QR estimator and establish that confidence regions based on the smoothed empirical likelihood ratio have coverage errors of order n^{-1} and may be Bartlett-corrected to produce regions with an error of order n^{-2}, where n denotes the sample size. We further extend these results to censored quantile regression models. Our results are extensions of the previous results of Chen and Hall (1993) to the regression contexts. Monte Carlo experiments suggest that the smoothed empirical likelihood confidence regions may be more accurate in small samples than the confidence regions that can be constructed from the smoothed bootstrap method recently suggested by Horowitz (1998).

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Bibliographic Info

Paper provided by Cowles Foundation for Research in Economics, Yale University in its series Cowles Foundation Discussion Papers with number 1453.

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Length: 44 pages
Date of creation: Mar 2004
Date of revision:
Handle: RePEc:cwl:cwldpp:1453

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Postal: Cowles Foundation, Yale University, Box 208281, New Haven, CT 06520-8281 USA

Related research

Keywords: Bartlett correction; Bootstrap; Edgeworth expansion; Empirical likelihood; Quantile regression model; Censored quantile regression model;

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References

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  1. Hahn, Jinyong, 1995. "Bootstrapping Quantile Regression Estimators," Econometric Theory, Cambridge University Press, vol. 11(01), pages 105-121, February.
  2. Hall, Peter & Horowitz, Joel L., 1990. "Bandwidth Selection in Semiparametric Estimation of Censored Linear Regression Models," Econometric Theory, Cambridge University Press, vol. 6(02), pages 123-150, June.
  3. Chen, Song Xi & Cui, Hengjian, 2007. "On the second-order properties of empirical likelihood with moment restrictions," Journal of Econometrics, Elsevier, vol. 141(2), pages 492-516, December.
  4. Whitney K. Newey & Richard J. Smith, 2004. "Higher Order Properties of Gmm and Generalized Empirical Likelihood Estimators," Econometrica, Econometric Society, vol. 72(1), pages 219-255, 01.
  5. Guido W. Imbens & Phillip Johnson & Richard H. Spady, 1995. "Information Theoretic Approaches to Inference in Moment Condition Models," NBER Technical Working Papers 0186, National Bureau of Economic Research, Inc.
  6. Buchinsky, Moshe, 1995. "Estimating the asymptotic covariance matrix for quantile regression models a Monte Carlo study," Journal of Econometrics, Elsevier, vol. 68(2), pages 303-338, August.
  7. Song Xi Chen & Hengjian Cui, 2006. "On Bartlett correction of empirical likelihood in the presence of nuisance parameters," Biometrika, Biometrika Trust, vol. 93(1), pages 215-220, March.
  8. Patrik Buggenberger & Richard Smith, 2003. "Generalized empirical likelihood estimators and tests under partial, weak and strong identification," CeMMAP working papers CWP08/03, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  9. Moshe Buchinsky, 1998. "Recent Advances in Quantile Regression Models: A Practical Guideline for Empirical Research," Journal of Human Resources, University of Wisconsin Press, vol. 33(1), pages 88-126.
  10. Koenker, Roger & Bassett, Gilbert, Jr, 1982. "Robust Tests for Heteroscedasticity Based on Regression Quantiles," Econometrica, Econometric Society, vol. 50(1), pages 43-61, January.
  11. Joel L. Horowitz, 1998. "Bootstrap Methods for Median Regression Models," Econometrica, Econometric Society, vol. 66(6), pages 1327-1352, November.
  12. Su, Liangjun & White, Halbert, 2003. "Testing Conditional Independence Via Empirical Likelihood," University of California at San Diego, Economics Working Paper Series qt35v8g0fm, Department of Economics, UC San Diego.
  13. repec:cup:etheor:v:11:y:1995:i:1:p:105-21 is not listed on IDEAS
  14. Koenker, Roger W & Bassett, Gilbert, Jr, 1978. "Regression Quantiles," Econometrica, Econometric Society, vol. 46(1), pages 33-50, January.
  15. Song Chen, 1993. "On the accuracy of empirical likelihood confidence regions for linear regression model," Annals of the Institute of Statistical Mathematics, Springer, vol. 45(4), pages 621-637, December.
  16. Donald, Stephen G. & Imbens, Guido W. & Newey, Whitney K., 2003. "Empirical likelihood estimation and consistent tests with conditional moment restrictions," Journal of Econometrics, Elsevier, vol. 117(1), pages 55-93, November.
  17. Powell, James L., 1984. "Least absolute deviations estimation for the censored regression model," Journal of Econometrics, Elsevier, vol. 25(3), pages 303-325, July.
  18. Francesco Bravo, 2002. "Testing linear restrictions in linear models with empirical likelihood," Econometrics Journal, Royal Economic Society, vol. 5(1), pages 104-130, June.
  19. Karun Adusumilli & Taisuke Otsu, 2014. "Empirical Likelihood for Random Sets," STICERD - Econometrics Paper Series /2014/574, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
  20. repec:cup:etheor:v:6:y:1990:i:2:p:123-50 is not listed on IDEAS
  21. Yuichi Kitamura, 2001. "Asymptotic Optimality of Empirical Likelihood for Testing Moment Restrictions," Econometrica, Econometric Society, vol. 69(6), pages 1661-1672, November.
  22. Bravo, Francesco, 2004. "Empirical Likelihood Based Inference With Applications To Some Econometric Models," Econometric Theory, Cambridge University Press, vol. 20(02), pages 231-264, April.
  23. Powell, James L., 1986. "Censored regression quantiles," Journal of Econometrics, Elsevier, vol. 32(1), pages 143-155, June.
  24. Horowitz, Joel L., 2001. "The Bootstrap," Handbook of Econometrics, in: J.J. Heckman & E.E. Leamer (ed.), Handbook of Econometrics, edition 1, volume 5, chapter 52, pages 3159-3228 Elsevier.
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Citations

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Cited by:
  1. Tang, Cheng Yong & Leng, Chenlei, 2012. "An empirical likelihood approach to quantile regression with auxiliary information," Statistics & Probability Letters, Elsevier, vol. 82(1), pages 29-36.
  2. Giuseppe Ragusa, 2011. "Minimum Divergence, Generalized Empirical Likelihoods, and Higher Order Expansions," Econometric Reviews, Taylor & Francis Journals, vol. 30(4), pages 406-456.
  3. Juan Carlos Escanciano & Carlos Velasco, 2008. "Specification Tests of Parametric Dynamic Conditional Quantiles," Caepr Working Papers 2008-021, Center for Applied Economics and Policy Research, Economics Department, Indiana University Bloomington.
  4. Wanrong Liu & Xuewen Lu, 2011. "Empirical likelihood for density-weighted average derivatives," Statistical Papers, Springer, vol. 52(2), pages 391-412, May.
  5. Pang, Lei & Lu, Wenbin & Wang, Huixia Judy, 2012. "Variance estimation in censored quantile regression via induced smoothing," Computational Statistics & Data Analysis, Elsevier, vol. 56(4), pages 785-796.
  6. Escanciano, J.C. & Goh, S.C., 2014. "Specification analysis of linear quantile models," Journal of Econometrics, Elsevier, vol. 178(P3), pages 495-507.
  7. Xiaofeng Lv & Rui Li, 2013. "Smoothed empirical likelihood analysis of partially linear quantile regression models with missing response variables," AStA Advances in Statistical Analysis, Springer, vol. 97(4), pages 317-347, October.
  8. Qi Li & Jeffrey Scott Racine, 2006. "Nonparametric Econometrics: Theory and Practice," Economics Books, Princeton University Press, edition 1, volume 1, number 8355.
  9. Kaplan, David M. & Sun, Yixiao, 2012. "Smoothed Estimating Equations For Instrumental Variables Quantile Regression," University of California at San Diego, Economics Working Paper Series qt888657tp, Department of Economics, UC San Diego.
  10. Otsu, Taisuke, 2008. "Conditional empirical likelihood estimation and inference for quantile regression models," Journal of Econometrics, Elsevier, vol. 142(1), pages 508-538, January.
  11. Philip Kostov, 2013. "Empirical likelihood estimation of the spatial quantile regression," Journal of Geographical Systems, Springer, vol. 15(1), pages 51-69, January.
  12. Parente, Paulo M.D.C. & Smith, Richard J., 2011. "Gel Methods For Nonsmooth Moment Indicators," Econometric Theory, Cambridge University Press, vol. 27(01), pages 74-113, February.

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