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Confidence Bands In Quantile Regression

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  • Härdle, Wolfgang K.
  • Song, Song

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.

Suggested Citation

  • Härdle, Wolfgang K. & Song, Song, 2010. "Confidence Bands In Quantile Regression," Econometric Theory, Cambridge University Press, vol. 26(4), pages 1180-1200, August.
    • Handle: RePEc:cup:etheor:v:26:y:2010:i:04:p:1180-1200_99
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    References listed on IDEAS

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    Cited by:

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    2. Marc Hallin & Zudi Lu & Davy Paindaveine & Miroslav Siman, 2012. "Local Constant and Local Bilinear Multiple-Output Quantile Regression," Working Papers ECARES ECARES 2012-003, ULB -- Universite Libre de Bruxelles.
    3. Wang, Weining & Bobojonov, Ihtiyor & Härdle, Wolfgang Karl & Odening, Martin, 2011. "Increasing weather risk: Fact of fiction?," SFB 649 Discussion Papers 2011-077, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    4. Song, Song & Ritov, Ya’acov & Härdle, Wolfgang K., 2012. "Bootstrap confidence bands and partial linear quantile regression," Journal of Multivariate Analysis, Elsevier, vol. 107(C), pages 244-262.
    5. Chao, Shih-Kang & Härdle, Wolfgang Karl & Wang, Weining, 2012. "Quantile regression in risk calibration," SFB 649 Discussion Papers 2012-006, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    6. Härdle, Wolfgang Karl & Ritov, Ya'acov & Wang, Weining, 2013. "Tie the straps: Uniform bootstrap con fidence bands for bounded influence curve estimators," SFB 649 Discussion Papers 2013-047, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    7. Härdle, Wolfgang Karl & Ritov, Ya’acov & Wang, Weining, 2015. "Tie the straps: Uniform bootstrap confidence bands for semiparametric additive models," Journal of Multivariate Analysis, Elsevier, vol. 134(C), pages 129-145.
    8. Zharova, Alona & Mihoci, Andrija & Härdle, Wolfgang Karl, 2016. "Academic ranking scales in economics: Prediction and imputation," SFB 649 Discussion Papers 2016-020, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    9. Haoze Hou & Wei Huang & Zheng Zhang, 2025. "Non-parametric Quantile Regression and Uniform Inference with Unknown Error Distribution," Papers 2504.01761, arXiv.org.
    10. Guo, Mengmeng & Zhou, Lhan & Huang, Jianhua Z. & Härdle, Wolfgang Karl, 2013. "Functional data analysis of generalized quantile regressions," SFB 649 Discussion Papers 2013-001, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    11. repec:hum:wpaper:sfb649dp2016-020 is not listed on IDEAS
    12. Curcio, Domenico & D’Amico, Simona & Gianfrancesco, Igor & Vioto, Davide, 2024. "Understanding the impact of the financial technology revolution on systemic risk: Evidence from US and EU diversified financials," Research in International Business and Finance, Elsevier, vol. 69(C).
    13. Härdle, Wolfgang Karl & Ritov, Ya'acov & Song, Song, 2010. "Partial linear quantile regression and bootstrap confidence bands," SFB 649 Discussion Papers 2010-002, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    14. Kim, Kun Ho & Chao, Shih-Kang & Härdle, Wolfgang Karl, 2020. "Simultaneous Inference of the Partially Linear Model with a Multivariate Unknown Function," IRTG 1792 Discussion Papers 2020-008, Humboldt University of Berlin, International Research Training Group 1792 "High Dimensional Nonstationary Time Series".
    15. Qu, Zhongjun & Yoon, Jungmo, 2015. "Nonparametric estimation and inference on conditional quantile processes," Journal of Econometrics, Elsevier, vol. 185(1), pages 1-19.
    16. Hasan, Iftekhar & Tunaru, Radu & Vioto, Davide, 2023. "Herding behavior and systemic risk in global stock markets," Journal of Empirical Finance, Elsevier, vol. 73(C), pages 107-133.
    17. Curcio, Domenico & Gianfrancesco, Igor & Onorato, Grazia & Vioto, Davide, 2024. "Do ESG scores affect financial systemic risk? Evidence from European banks and insurers," Research in International Business and Finance, Elsevier, vol. 69(C).
    18. Katharina Proksch, 2016. "On confidence bands for multivariate nonparametric regression," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 68(1), pages 209-236, February.
    19. Duygun, Meryem & Tunaru, Radu & Vioto, Davide, 2021. "Herding by corporates in the US and the Eurozone through different market conditions," Journal of International Money and Finance, Elsevier, vol. 110(C).
    20. repec:hum:wpaper:sfb649dp2014-028 is not listed on IDEAS
    21. Toshio Honda, 2013. "Nonparametric quantile regression with heavy-tailed and strongly dependent errors," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 65(1), pages 23-47, February.
    22. Ali Al-Sharadqah & Majid Mojirsheibani, 2020. "A simple approach to construct confidence bands for a regression function with incomplete data," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 104(1), pages 81-99, March.
    23. repec:hum:wpaper:sfb649dp2011-077 is not listed on IDEAS
    24. repec:hum:wpaper:sfb649dp2013-001 is not listed on IDEAS
    25. Härdle, Wolfgang Karl & Spokoiny, Vladimir & Wang, Weining, 2010. "Local quantile regression," SFB 649 Discussion Papers 2011-005, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.

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