Finite-sample Distribution-free Inference in Linear Median Regression under Heteroskedasticity and Nonlinear Dependence of Unknown Form

We construct finite-sample distribution-free tests and confidence sets for the parametersof a linear median regression where no parametric assumptions are imposed on thenoise distribution. The setup we consider allows for nonnormality, heteroskedasticityand nonlinear serial dependence in the errors. Such semiparametric models are usuallyanalyzed using only asymptotically justified approximate methods, which can be arbitrarilyunreliable in finite samples. We consider first the property of mediangale – themedian-based analogue of a martingale difference – and show that the signs of mediangalesequences follow a nuisance-parameter free-distribution despite the presence ofnonlinear dependence and heterogeneity of unknown form. We point out that a simultaneousinference approach in conjunction with sign transformations do provide statisticswith the required pivotality features – in addition to usual robustness properties. Thosesign-based statistics are exploited – usingMonte Carlo tests and projection techniques –in order to produce valid inference in finite samples. An asymptotic theory which holdsunder weaker assumptions is also provided. Finally, simulation results illustrating theperformance and two applications are presented.