Kernel-weighted specification testing under general distributions

Kernel-weighted test statistics have been widely used in a variety of settings including non-stationary regression, survival analysis, propensity score and panel data models. We develop the limit theory for a kernel-weighted specification test of a parametric conditional mean when the law of the regressors may not be absolutely continuous to the Lebesgue measure and admits non-trivial singular components. In the special case of absolutely continuous measures, our approach weakens the usual regularity conditions. This result is of independent interest and may be useful in other applications that utilize kernel smoothed statistics. Simulations illustrate the non-trivial impact of the distribution of the conditioning variables on the power properties of the test statistic.