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Finite-Sample Generalized Confidence Distributions and Sign-Based Robust Estimators in Median Regressions with Heterogeneous Dependent Errors

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  • Élise, COUDIN
  • Jean-Marie DUFOUR

Abstract

We study the problem of estimating the parameters of a linear median regression without any assumption on the shape of the error distribution – including no condition on the existence of moments – allowing for heterogeneity (or heteroskedasticity) of unknown form, noncontinuous distributions, and very general serial dependence (linear and nonlinear). This is done through a reverse inference approach, based on a distribution-free testing theory [Coudin and Dufour (2009, The Econometrics Journal)], from which confidence sets and point estimators are subsequently generated. The estimation problem is tackled in two complementary ways. First, we show how confidence distributions for model parameters can be applied in such a context. Such distributions – which can be interpreted as a form of fiducial inference – provide a frequency-based method for associating probabilities with subsets of the parameter space (like posterior distributions do in a Bayesian setup) without the introduction of prior distributions. We consider generalized confidence distributions applicable to multidimensional parameters, and we suggest the use of a projection technique for confidence inference on individual model parameters. Second, we propose point estimators, which have a natural association with confidence distributions. These estimators are based on maximizing test p-values and inherit robustness properties from the generating distribution-free tests. Both finite-sample and large-sample properties of the proposed estimators are established under weak regularity conditions. We show they are median unbiased (under symmetry and estimator unicity) and possess equivariance properties. Consistency and asymptotic normality are established without any moment existence assumption on the errors, allowing for noncontinuous distributions, heterogeneity and serial dependence of unknown form. These conditions are considerably weaker than those used to show corresponding results for LAD estimators. In a Monte Carlo study of bias and RMSE, we show sign-based estimators perform better than LAD-type estimators in heteroskedastic settings. We present two empirical applications, which involve financial and macroeconomic data, both affected by heavy tails (non-normality) and heteroskedasticity: a trend model for the S&P index, and an equation used to study β-convergence of ouput levels across U.S. States.

Suggested Citation

  • Élise, COUDIN & Jean-Marie DUFOUR, 2017. "Finite-Sample Generalized Confidence Distributions and Sign-Based Robust Estimators in Median Regressions with Heterogeneous Dependent Errors," Cahiers de recherche 01-2017, Centre interuniversitaire de recherche en économie quantitative, CIREQ.
  • Handle: RePEc:mtl:montec:01-2017
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    References listed on IDEAS

    as
    1. Jean-Marie Dufour & Mohamed Taamouti, 2005. "Projection-Based Statistical Inference in Linear Structural Models with Possibly Weak Instruments," Econometrica, Econometric Society, vol. 73(4), pages 1351-1365, July.
    2. 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.
    3. Zhao, Quanshui, 2001. "Asymptotically Efficient Median Regression In The Presence Of Heteroskedasticity Of Unknown Form," Econometric Theory, Cambridge University Press, vol. 17(04), pages 765-784, August.
    4. Weiss, Andrew A., 1991. "Estimating Nonlinear Dynamic Models Using Least Absolute Error Estimation," Econometric Theory, Cambridge University Press, vol. 7(01), pages 46-68, March.
    5. Elise Coudin & Jean-Marie Dufour, 2007. "Finite-sample Distribution-free Inference in Linear Median Regression under Heteroskedasticity and Nonlinear Dependence of Unknown Form," Working Papers 2007-38, Center for Research in Economics and Statistics.
    6. 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.
    7. Bantli, Faouzi El & Hallin, Marc, 1999. "L1-estimation in linear models with heterogeneous white noise," Statistics & Probability Letters, Elsevier, vol. 45(4), pages 305-315, December.
    8. 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.
    9. Jean-Marie Dufour, 1997. "Some Impossibility Theorems in Econometrics with Applications to Structural and Dynamic Models," Econometrica, Econometric Society, vol. 65(6), pages 1365-1388, November.
    10. Jean-Marie Dufour & Jan F. Kiviet, 1998. "Exact Inference Methods for First-Order Autoregressive Distributed Lag Models," Econometrica, Econometric Society, vol. 66(1), pages 79-104, January.
    11. Dufour, Jean-Marie & Jasiak, Joann, 2001. "Finite Sample Limited Information Inference Methods for Structural Equations and Models with Generated Regressors," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 42(3), pages 815-843, August.
    12. Dufour, Jean-Marie, 2006. "Monte Carlo tests with nuisance parameters: A general approach to finite-sample inference and nonstandard asymptotics," Journal of Econometrics, Elsevier, vol. 133(2), pages 443-477, August.
    13. 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.
    14. Koenker,Roger, 2005. "Quantile Regression," Cambridge Books, Cambridge University Press, number 9780521845731, October.
    15. Weiss, Andrew A., 1990. "Least absolute error estimation in the presence of serial correlation," Journal of Econometrics, Elsevier, vol. 44(1-2), pages 127-158.
    16. Dufour, Jean-Marie & Valéry, Pascale, 2009. "Exact and asymptotic tests for possibly non-regular hypotheses on stochastic volatility models," Journal of Econometrics, Elsevier, vol. 150(2), pages 193-206, June.
    17. Dufour, Jean-Marie, 1990. "Exact Tests and Confidence Sets in Linear Regressions with Autocorrelated Errors," Econometrica, Econometric Society, vol. 58(2), pages 475-494, March.
    18. Phillips, P.C.B., 1991. "A Shortcut to LAD Estimator Asymptotics," Econometric Theory, Cambridge University Press, vol. 7(04), pages 450-463, December.
    19. Pollard, David, 1991. "Asymptotics for Least Absolute Deviation Regression Estimators," Econometric Theory, Cambridge University Press, vol. 7(02), pages 186-199, June.
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    More about this item

    Keywords

    sign-based methods; median regression; test inversion; Hodges-Lehmann estimators; confidence distributions; p-value function; least absolute deviation estimators; quantile regressions; sign test; simultaneous inference; Monte Carlo tests; projection methods; non-normality; heteroskedasticity; serial dependence; GARCH; stochastic volatility;

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • C15 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Statistical Simulation Methods: General

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