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 output levels across U.S. States.

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