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