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.

翻译：核加权检验统计量已广泛应用于非平稳回归、倾向得分推断及面板数据模型等多种场景。本文针对回归变量分布可能不绝对连续于勒贝格测度且包含奇异成分的情况，建立了参数条件均值核基设定检验的极限理论。该结果具有独立学术价值，可推广至其他采用核平滑U统计量的应用场景。模拟实验表明，条件变量分布对检验统计量的势特性具有不可忽略的影响。