Finite-sample generalized confidence distributions and sign-based robust estimators in median regressions with heterogeneous dependent errors

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 sign-based testing theory, from which confidence sets and point estimators are subsequently generated. 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 that 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. A Monte Carlo study of bias and RMSE shows sign-based estimators perform better than LAD-type estimators in various heteroskedastic settings. We illustrate the use of sign-based estimators on an example of β-convergence of output levels across U.S. states.

Keywords:

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

JEL Classification:

• C13
• C12
• C14
• C15

Acknowledgments

The authors thank Magali Beffy, Marine Carrasco, Frédéric Jouneau, Marc Hallin, Thierry Magnac, Bill McCausland, Benoit Perron, and Alain Trognon for useful comments and constructive discussions. Earlier versions of this paper were presented at the 2005 Econometric Society World Congress (London), the Econometric Society European Meeting 2007 (Budapest), CREST (Paris), the Conference on Nonparametric statistics and time series in honor of Marc Hallin (Bruxelles, December 2009), and the First French Econometrics Conference Celebrating Alain Monfort (Toulouse, December 2009).

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 This holds also for quantile regressions (Koenker 2005; Koenker and Bassett 1978), which can be viewed as extensions of median regression.

2 See also Dwass (1957), Barnard (1963), Dufour (1990, 1997), Dufour and Kiviet (1998), Abdelkhalek and Dufour (1998), Dufour and Jasiak (2001), Dufour and Taamouti (2005). And for an alternative finite-sample inference exploiting a quantile version of the same sign pivotality result, which holds if the observations are X-conditionally independent, see Chernozhukov, Hansen, and Jansson (2009).

3 For continuous distributions, just note that $P_{\beta }[\beta \le C{D}^{-1}(\alpha )]=P_{\beta }\{CD(\beta )\le CD(C{D}^{-1}(\alpha ))\}=P_{\beta }\{CD(\beta )\le \alpha ]\}=\alpha .$ Schweder and Hjort (2002) introduce the notion of “degree of confidence” $CD({\beta }_0)$ of the statement $\beta \le {\beta }_0$ which is equals to the p-value of a test $\beta \le {\beta }_0$ versus the alternative $\beta >{\beta }_0.$

4 Those relations are stated in Lemma 2 of Schweder and Hjort (2002): the confidence of the statement “$\beta \le {\beta }_0$” is the degree of confidence $CD({\beta }_0)$ for the confidence interval $(-\infty ,C{D}^{-1}(CD({\beta }_0))]$, and is equal to the p-value of a test of $H_0^{\ast }({\beta }_0):\beta \le {\beta }_0$ vs. $H_1^{\ast }({\beta }_0):\beta >{\beta }_0.$

5 Continuous uniform distribution is obtained using a randomization process on ties in Coudin and Dufour (2009).

6 Due to the use of the nonlinear p-value transformation (along with the associated finite-sample distributional theory), the GMM interpretation does not stricto sensu generally hold, except possibly through an asymptotic equivalence.

7 To see that it is a union of convex sets just remark that the reciprocal image of n fixed signs is convex.

8 Assumption 6.4 can be slightly relaxed covering error terms with mass point if the objective function involves randomized signs instead of usual signs.

9 See Fitzenberger (1998) for the derivation of the LAD asymptotics in a similar setup, and Bassett and Koenker (1978) or Weiss (1991) for a derivation of the LAD asymptotics under sign independence.

Additional information

Funding

This work was supported by the William Dow Chair in Political Economy (McGill University), the Bank of Canada (Research Fellowship), the Toulouse School of Economics (Pierre-de-Fermat Chair of excellence), the Universitad Carlos III de Madrid (Banco Santander de Madrid Chair of excellence), a Guggenheim Fellowship, a Konrad-Adenauer Fellowship (Alexander-von-Humboldt Foundation, Germany), the Canadian Network of Centres of Excellence [program on Mathematics of Information Technology and Complex Systems (MITACS)], the Natural Sciences and Engineering Research Council of Canada, the Social Sciences and Humanities Research Council of Canada, and the Fonds de recherche sur la société et la culture (Québec).