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Asymptotically Efficient Median Regression In The Presence Of Heteroskedasticity Of Unknown Form

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  • Zhao, Quanshui

Abstract

We consider a linear model with heteroskedasticity of unknown form. Using Stone's (1977, Annals of Statistics 5, 595–645) k nearest neighbors (k-NN) estimation approach, the optimal weightings for efficient least absolute deviation regression are estimated consistently using residuals from preliminary estimation. The reweighted least absolute deviation or median regression estimator with the estimated weights is shown to be equivalent to the estimator using the true but unknown weights under mild conditions. Asymptotic normality of the estimators is also established. In the finite sample case, the proposed estimators are found to outperform the generalized least squares method of Robinson (1987, Econometrica 55, 875–891) and the one-step estimator of Newey and Powell (1990, Econometric Theory 6, 295–317) based on a Monte Carlo simulation experiment.

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  • Zhao, Quanshui, 2001. "Asymptotically Efficient Median Regression In The Presence Of Heteroskedasticity Of Unknown Form," Econometric Theory, Cambridge University Press, vol. 17(4), pages 765-784, August.
    • Handle: RePEc:cup:etheor:v:17:y:2001:i:04:p:765-784_17
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    Cited by:

    1. Ma, Lingjie & Koenker, Roger, 2006. "Quantile regression methods for recursive structural equation models," Journal of Econometrics, Elsevier, vol. 134(2), pages 471-506, October.
    2. Javier Alejo & Antonio F Galvao & Gabriel Montes-Rojas, 2023. "A first-stage representation for instrumental variables quantile regression," The Econometrics Journal, Royal Economic Society, vol. 26(3), pages 350-377.
    3. Elise COUDIN, Jean-Marie DUFOUR, 2008. "Hodges-Lehmann Sign-based Estimators and Generalized Confidence Distributions in Linear Median Regressions with Moment-free Heterogenous Errors and Dependence of Unknown Form," Working Papers 2008-33, Center for Research in Economics and Statistics.
    4. Wilk, M. & Zaigraev, A., 2017. "DS-optimal designs for random coefficient first-degree regression model with heteroscedastic errors," Statistics & Probability Letters, Elsevier, vol. 128(C), pages 28-34.
    5. Hallin, M. & Vermandele, C. & Werker, B.J.M., 2003. "Serial and Nonserial Sign-and-Rank Statistics : Asymptotic Representation and Asymptotic Normality," Other publications TiSEM 620d09ba-f476-426d-b236-3, Tilburg University, School of Economics and Management.
    6. Ke Zhu, 2018. "Statistical inference for autoregressive models under heteroscedasticity of unknown form," Papers 1804.02348, arXiv.org, revised Aug 2018.
    7. Wu Wang & Zhongyi Zhu, 2017. "Conditional empirical likelihood for quantile regression models," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 80(1), pages 1-16, January.
    8. Elise Coudin & Jean-Marie Dufour, 2017. "Finite-sample generalized confidence distributions and sign-based robust estimators in median regressions with heterogenous dependent errors," CIRANO Working Papers 2017s-06, CIRANO.
    9. Marilena Furno, 2012. "Tests for structural break in quantile regressions," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 96(4), pages 493-515, October.
    10. Lingjie Ma & Larry Pohlman, 2008. "Return forecasts and optimal portfolio construction: a quantile regression approach," The European Journal of Finance, Taylor & Francis Journals, vol. 14(5), pages 409-425.
    11. Lee, Dong Jin & Kim, Tae-Hwan & Mizen, Paul, 2021. "Impulse response analysis in conditional quantile models with an application to monetary policy," Journal of Economic Dynamics and Control, Elsevier, vol. 127(C).
    12. Otsu, Taisuke, 2008. "Conditional empirical likelihood estimation and inference for quantile regression models," Journal of Econometrics, Elsevier, vol. 142(1), pages 508-538, January.
    13. Komunjer, Ivana & Vuong, Quang, 2010. "Efficient estimation in dynamic conditional quantile models," Journal of Econometrics, Elsevier, vol. 157(2), pages 272-285, August.
    14. Sweeney, Stuart & Davenport, Frank & Grace, Kathryn, 2013. "Combining insights from quantile and ordinal regression: Child malnutrition in Guatemala," Economics & Human Biology, Elsevier, vol. 11(2), pages 164-177.
    15. Oberhofer, Walter & Haupt, Harry, 2003. "Nonlinear quantile regression under dependence and heterogeneity," University of Regensburg Working Papers in Business, Economics and Management Information Systems 388, University of Regensburg, Department of Economics.
    16. Chen, Tao & Parker, Thomas, 2014. "Semiparametric efficiency for partially linear single-index regression models," Journal of Multivariate Analysis, Elsevier, vol. 130(C), pages 376-386.
    17. Lingjie Ma & Roger Koenker, 2004. "Quantile regression methods for recursive structural equation models," CeMMAP working papers 01/04, Institute for Fiscal Studies.
    18. Komunjer, Ivana, 2013. "Quantile Prediction," Handbook of Economic Forecasting, in: G. Elliott & C. Granger & A. Timmermann (ed.), Handbook of Economic Forecasting, edition 1, volume 2, chapter 0, pages 961-994, Elsevier.
    19. He X. & Zhu L-X., 2003. "A Lack-of-Fit Test for Quantile Regression," Journal of the American Statistical Association, American Statistical Association, vol. 98, pages 1013-1022, January.

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