Minimum Contrast Empirical Likelihood Inference of Discontinuity in Density*

Abstract

This article investigates the asymptotic properties of a simple empirical-likelihood-based inference method for discontinuity in density. The parameter of interest is a function of two one-sided limits of the probability density function at (possibly) two cut-off points. Our approach is based on the first-order conditions from a minimum contrast problem. We investigate both first-order and second-order properties of the proposed method. We characterize the leading coverage error of our inference method and propose a coverage-error-optimal (CE-optimal, hereafter) bandwidth selector. We show that the empirical likelihood ratio statistic is Bartlett correctable. An important special case is the manipulation testing problem in a regression discontinuity design (RDD), where the parameter of interest is the density difference at a known threshold. In RDD, the continuity of the density of the assignment variable at the threshold is considered as a “no-manipulation” behavioral assumption, which is a testable implication of an identifying condition for the local average treatment effect. When specialized to the manipulation testing problem, the CE-optimal bandwidth selector has an explicit form. We propose a data-driven CE-optimal bandwidth selector for use in practice. Results from Monte Carlo simulations are presented. Usefulness of our method is illustrated by an empirical example.