Generic Inference on Quantile and Quantile Effect Functions for Discrete Outcomes

Quantile and quantile effect functions are important tools for descriptive and inferential analysis due to their natural and intuitive interpretation. Existing inference methods for these functions do not apply to discrete and mixed continuous-discrete random variables. This paper offers a simple, practical construction of the simultaneous confidence bands for quantile and quantile effect functions. It is based on a natural transformation of simultaneous confidence bands for distribution functions, which are readily available for many problems. The construction is generic and does not depend on the nature of the underlying problem. It works in conjunction with parametric, semiparametric, and nonparametric modeling strategies and does not depend on the sampling scheme. We apply our method to characterize the distributional impact of insurance coverage on health care utilization and obtain the distributional decomposition of the racial test score gap. Our analysis generates new, interesting empirical findings, and complements previous analyses that focused on mean effects only. In both applications, the outcomes of interest are discrete rendering existing inference methods invalid for obtaining uniform confidence bands for quantile and quantile effects functions.