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On monotonicity of regression quantile functions

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  • Neocleous, Tereza
  • Portnoy, Stephen

Abstract

In the linear regression quantile model, the conditional quantile of the response, Y, given x is QYx([tau])[reverse not equivalent]x'[beta]([tau]). Though QYx([tau]) must be monotonically increasing, the Koenker-Bassett regression quantile estimator, , is not monotonic outside a vanishingly small neighborhood of . Given a grid of mesh [delta]n, let be the linear interpolation of the values of along the grid. We show here that for a range of rates, [delta]n, will be strictly monotonic (with probability tending to one) and will be asymptotically equivalent to in the sense that n1/2 times the difference tends to zero at a rate depending on [delta]n.

Suggested Citation

  • Neocleous, Tereza & Portnoy, Stephen, 2008. "On monotonicity of regression quantile functions," Statistics & Probability Letters, Elsevier, vol. 78(10), pages 1226-1229, August.
    • Handle: RePEc:eee:stapro:v:78:y:2008:i:10:p:1226-1229
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    References listed on IDEAS

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    1. Koenker,Roger, 2005. "Quantile Regression," Cambridge Books, Cambridge University Press, number 9780521845731, July.
    2. Koenker, Roger W & Bassett, Gilbert, Jr, 1978. "Regression Quantiles," Econometrica, Econometric Society, vol. 46(1), pages 33-50, January.
    3. Portnoy, Stephen, 1991. "Asymptotic behavior of regression quantiles in non-stationary, dependent cases," Journal of Multivariate Analysis, Elsevier, vol. 38(1), pages 100-113, July.
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    Cited by:

    1. Yufeng Liu & Yichao Wu, 2011. "Simultaneous multiple non-crossing quantile regression estimation using kernel constraints," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 23(2), pages 415-437.
    2. Qu, Zhongjun & Yoon, Jungmo, 2015. "Nonparametric estimation and inference on conditional quantile processes," Journal of Econometrics, Elsevier, vol. 185(1), pages 1-19.
    3. Peng, Limin, 2012. "Self-consistent estimation of censored quantile regression," Journal of Multivariate Analysis, Elsevier, vol. 105(1), pages 368-379.
    4. Christis Katsouris, 2022. "Asymptotic Theory for Unit Root Moderate Deviations in Quantile Autoregressions and Predictive Regressions," Papers 2204.02073, arXiv.org, revised Aug 2023.
    5. Victor Chernozhukov & Iván Fernández-Val & Blaise Melly, 2022. "Fast algorithms for the quantile regression process," Empirical Economics, Springer, vol. 62(1), pages 7-33, January.
    6. Yijian Huang, 2017. "Restoration of Monotonicity Respecting in Dynamic Regression," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(518), pages 613-622, April.
    7. Antonio F. Galvao JR. & Gabriel Montes-Rojas & Sung Y. Park, 2013. "Quantile Autoregressive Distributed Lag Model with an Application to House Price Returns," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 75(2), pages 307-321, April.
    8. Stephen Portnoy & Guixian Lin, 2010. "Asymptotics for censored regression quantiles," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 22(1), pages 115-130.
    9. Juban, Romain & Ohlsson, Henrik & Maasoumy, Mehdi & Poirier, Louis & Kolter, J. Zico, 2016. "A multiple quantile regression approach to the wind, solar, and price tracks of GEFCom2014," International Journal of Forecasting, Elsevier, vol. 32(3), pages 1094-1102.

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