Specification tests for the distribution of errors in nonparametric regression: a martingale approach

Abstract

We discuss how to test whether the distribution of regression errors belongs to a parametric family of continuous distribution functions, making no parametric assumption about the conditional mean or the conditional variance in the regression model. We propose using test statistics that are based on a martingale transform of the estimated empirical process. We prove that the resulting test statistics are asymptotically distribution-free, and a set of Monte Carlo experiments shows that they work reasonably well in practice.