Testing Continuity of a Density via g-order statistics in the Regression Discontinuity Design

TL;DR

This paper introduces an approximate sign test based on g-order statistics to assess the continuity of a density at a cutoff in regression discontinuity design, offering a simple, valid, and robust alternative to existing methods.

Contribution

It proposes a novel approximate sign test for density continuity in RDD, valid under weaker conditions and applicable in two asymptotic frameworks, with demonstrated finite sample performance.

Findings

Test maintains nominal significance level asymptotically.

Method performs well in simulations, controlling false positives.

Applied successfully to a real RDD example in political science.

Abstract

In the regression discontinuity design (RDD), it is common practice to assess the credibility of the design by testing the continuity of the density of the running variable at the cut-off, e.g., McCrary (2008). In this paper we propose an approximate sign test for continuity of a density at a point based on the so-called g-order statistics, and study its properties under two complementary asymptotic frameworks. In the first asymptotic framework, the number q of observations local to the cut-off is fixed as the sample size n diverges to infinity, while in the second framework q diverges to infinity slowly as n diverges to infinity. Under both of these frameworks, we show that the test we propose is asymptotically valid in the sense that it has limiting rejection probability under the null hypothesis not exceeding the nominal level. More importantly, the test is easy to implement, asymptotically valid under weaker conditions than those used by competing methods, and exhibits finite sample validity under stronger conditions than those needed for its asymptotic validity. In a simulation study, we find that the approximate sign test provides good control of the rejection probability under the null hypothesis while remaining competitive under the alternative hypothesis. We finally apply our test to the design in Lee (2008), a well-known application of the RDD to study incumbency advantage.