Manipulation of the running variable in the regression discontinuity design: A density test

Abstract

Standard sufficient conditions for identification in the regression discontinuity design are continuity of the conditional expectation of counterfactual outcomes in the running variable. These continuity assumptions may not be plausible if agents are able to manipulate the running variable. This paper develops a test of manipulation related to continuity of the running variable density function. The methodology is applied to popular elections to the House of Representatives, where sorting is neither expected nor found, and to roll-call voting in the House, where sorting is both expected and found. * I thank two anonymous referees for comments, the editors for multiple suggestions that substantially improved the paper, Jack Porter, John DiNardo, and Serena Ng for discussion, Jonah Gelbach for computing improvements, and Ming-Yen Cheng for manuscripts. Any errors are my own.

Replacing churches and mason lodges? Tax exemptions and rural development

“…Moreover, Figure 4 shows no discontinuity in the density of the forcing variable at the threshold. This is consistent with the fact that areas had no way to manipulate their value of the forcing variable (McCrary, 2008). 12 equal to 1 if the area satisfies the population density criterion.…”

Improving College Access and Success for Low-Income Students: Evidence from a Large Need-Based Grant Program

“…The x-axis is the relative distance between the applicant's parental income and the income eligibility threshold. The optimal bandwidth and bin size are obtained using the selection procedure proposed by McCrary (2008). Relative Income−Distance to Eligibility Cutoff Notes: The circles represent the average fraction of applicants who were awarded a conditional grant (left panels) and the average amount of allowance awarded (right panels) per interval of relative income-distance to the eligibility thresholds.…”

“…To check for the absence of manipulation of the forcing variables, we start by examining whether the density of parental income is continuous at the eligibility cuto↵s using the test developed by McCrary (2008). This test, which is performed for each treatment sample separately, consists of running kernel local linear regressions of the log of the density separately on both sides of the thresholds.…”

“…The table rows show the estimated discontinuity in the density function of the assignment variable at the threshold, its standard error (in parenthesis), the associated z-statistic, the estimated optimal bandwidth, bin size and the number of observations. The optimal bandwidth and bin size are obtained using the selection procedure proposed by McCrary (2008). Z-test values lower than 1.64 suggest no statistical evidence of a discontinuity in the density of the assignment variable at the income eligibility thresholds.…”

Assessing the effects of disability insurance experience rating. The case of The Netherlands

“…In particular, there is no evidence that the wage costs of rms concentrate just below the threshold value. We have also formally tested this with the discontinuity test that is suggested by McCrary (2008). The null hypothesis of a continuous wage sum around the threshold could not be rejected for any year Third, our estimation strategy assumes that rms are classi ed as small or large over a longer stretch of time.…”