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Finite-sample distribution-free inference in linear median regressions under heteroscedasticity and non-linear dependence of unknown form

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  • Elise Coudin
  • Jean-Marie Dufour

Abstract

We construct finite-sample distribution-free tests and confidence sets for the parameters of a linear median regression, where no parametric assumption is imposed on the noise distribution. The set-up studied allows for non-normality, heteroscedasticity, non-linear serial dependence of unknown forms as well as for discrete distributions. We consider a mediangale structure--the median-based analogue of a martingale difference--and show that the signs of mediangale sequences follow a nuisance-parameter-free distribution despite the presence of non-linear dependence and heterogeneity of unknown form. We point out that a simultaneous inference approach in conjunction with sign transformations yield statistics with the required pivotality features--in addition to usual robustness properties. Monte Carlo tests and projection techniques are then exploited to produce finite-sample tests and confidence sets. Further, under weaker assumptions, which allow for weakly exogenous regressors and a wide class of linear dependence schemes in the errors, we show that the procedures proposed remain asymptotically valid. The regularity assumptions used are notably less restrictive than those required by procedures based on least absolute deviations (LAD). Simulation results illustrate the performance of the procedures. Finally, the proposed methods are applied to tests of the drift in the Standard and Poor's composite price index series (allowing for conditional heteroscedasticity of unknown form). Copyright (C) The Author(s). Journal compilation (C) Royal Economic Society 2009

Suggested Citation

  • Elise Coudin & Jean-Marie Dufour, 2009. "Finite-sample distribution-free inference in linear median regressions under heteroscedasticity and non-linear dependence of unknown form," Econometrics Journal, Royal Economic Society, vol. 12(s1), pages 19-49, January.
  • Handle: RePEc:ect:emjrnl:v:12:y:2009:i:s1:p:s19-s49
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    1. Touhami Abdelkhalek & Jean-Marie Dufour, 1998. "Statistical Inference For Computable General Equilibrium Models, With Application To A Model Of The Moroccan Economy," The Review of Economics and Statistics, MIT Press, vol. 80(4), pages 520-534, November.
    2. Xiaohong Chen & Oliver Linton & Ingrid Van Keilegom, 2003. "Estimation of Semiparametric Models when the Criterion Function Is Not Smooth," Econometrica, Econometric Society, vol. 71(5), pages 1591-1608, September.
    3. Campbell, Bryan & Dufour, Jean-Marie, 1997. "Exact Nonparametric Tests of Orthogonality and Random Walk in the Presence of a Drift Parameter," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 38(1), pages 151-173, February.
    4. 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.
    5. Andrews, Donald W K, 1991. "Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation," Econometrica, Econometric Society, vol. 59(3), pages 817-858, May.
    6. Moshe Buchinsky & Jinyong Hahn, 1998. "An Alternative Estimator for the Censored Quantile Regression Model," Econometrica, Econometric Society, vol. 66(3), pages 653-672, May.
    7. Campbell, Bryan & Dufour, Jean-Marie, 1991. "Over-rejections in rational expectations models : A non-parametric approach to the Mankiw-Shapiro problem," Economics Letters, Elsevier, vol. 35(3), pages 285-290, March.
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    Cited by:

    1. Kim, Jihyun & Meddahi, Nour, 2020. "Volatility Regressions with Fat Tails," TSE Working Papers 20-1097, Toulouse School of Economics (TSE).
    2. Bertanha, Marinho & Moreira, Marcelo J., 2020. "Impossible inference in econometrics: Theory and applications," Journal of Econometrics, Elsevier, vol. 218(2), pages 247-270.
    3. repec:hal:journl:peer-00834424 is not listed on IDEAS
    4. Kim, Jihyun & Meddahi, Nour, 2020. "Volatility regressions with fat tails," Journal of Econometrics, Elsevier, vol. 218(2), pages 690-713.
    5. Khalaf, Lynda & Saunders, Charles J., 2017. "Monte Carlo forecast evaluation with persistent data," International Journal of Forecasting, Elsevier, vol. 33(1), pages 1-10.
    6. 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.
    7. Gungor, Sermin & Luger, Richard, 2020. "Small-sample tests for stock return predictability with possibly non-stationary regressors and GARCH-type effects," Journal of Econometrics, Elsevier, vol. 218(2), pages 750-770.
    8. Kaveh Salehzadeh Nobari, 2021. "Pair copula constructions of point-optimal sign-based tests for predictive linear and nonlinear regressions," Papers 2111.04919, arXiv.org.
    9. Hallin, M. & van den Akker, R. & Werker, B.J.M., 2011. "A Class of Simple Distribution-free Rank-based Unit Root Tests (Revision of DP 2010-72)," Other publications TiSEM 004c9726-ec6a-4884-8238-d, Tilburg University, School of Economics and Management.
    10. Dufour, Jean-Marie & Taamouti, Abderrahim, 2008. "Exact optimal and adaptive inference in regression models under heteroskedasticity and non-normality of unknown forms," UC3M Working papers. Economics we086027, Universidad Carlos III de Madrid. Departamento de Economía.
    11. Beaulieu, Marie-Claude & Dufour, Jean-Marie & Khalaf, Lynda, 2010. "Asset-pricing anomalies and spanning: Multivariate and multifactor tests with heavy-tailed distributions," Journal of Empirical Finance, Elsevier, vol. 17(4), pages 763-782, September.
    12. Jihyun Kim & Nour Meddahi, 2020. "Volatility Regressions with Fat Tails," Post-Print hal-03142647, HAL.
    13. Luger, Richard, 2012. "Finite-sample bootstrap inference in GARCH models with heavy-tailed innovations," Computational Statistics & Data Analysis, Elsevier, vol. 56(11), pages 3198-3211.
    14. Elise Coudin & Jean-Marie Dufour, 2010. "Finite and Large Sample Distribution-Free Inference in Median Regressions with Instrumental Variables," Working Papers 2010-56, Center for Research in Economics and Statistics.
    15. Alex Maynard, 2006. "The forward premium anomaly: statistical artefact or economic puzzle? New evidence from robust tests," Canadian Journal of Economics, Canadian Economics Association, vol. 39(4), pages 1244-1281, November.
    16. Dufour, Jean-Marie & Taamouti, Abderrahim, 2010. "Exact optimal inference in regression models under heteroskedasticity and non-normality of unknown form," Computational Statistics & Data Analysis, Elsevier, vol. 54(11), pages 2532-2553, November.
    17. Hallin, Marc & van den Akker, Ramon & Werker, Bas J.M., 2011. "A class of simple distribution-free rank-based unit root tests," Journal of Econometrics, Elsevier, vol. 163(2), pages 200-214, August.
    18. 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.
    19. Gossner, Olivier & Schlag, Karl H., 2013. "Finite-sample exact tests for linear regressions with bounded dependent variables," Journal of Econometrics, Elsevier, vol. 177(1), pages 75-84.
    20. Chen, Min & Zhu, Ke, 2014. "Sign-based specification tests for martingale difference with conditional heteroscedasity," MPRA Paper 56347, University Library of Munich, Germany.
    21. Oliver Gossner & Karl Schlag, 2012. "Finite Sample Exact tests for Linear," Vienna Economics Papers 1201, University of Vienna, Department of Economics.

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