A Nonparametric Test for Testing Heterogeneity in Conditional Quantile Treatment Effects

This paper proposes a novel test to assess whether there exists heterogeneously distributional effect for an intervention on outcome of interest across different sub-populations defined by covariates of interest. Specifically, we develop a nonparametric test, a consistent test statistic based on the Cramer-von Mises type criterion, which is for the null hypothesis that the treatment has a constant quantile effect for all subpopulations defined by covariates of interest. Under some regular conditions, we establish the asymptotic distribution of the proposed test statistic under the null hypothesis and its consistency against fixed alternatives, together with studying the power of our test against a sequence of local alternatives. Furthermore, a nonparametric Bootstrap procedure is suggested to approximate the finite-sample null distribution of the proposed test and the asymptotic validity of the proposed Bootstrap test is also established. Through Monte Carlo simulations, we demonstrate the power properties of the test in finite samples. Finally, the proposed testing approach is applied to investigating whether there exists heterogeneity for the quantile treatment effect of maternal smoking during pregnancy on infant birth weight across different age groups of mothers.