Honest inference for discrete outcomes in regression discontinuity designs

We investigate the consequences of discreteness in the assignment variable in regression-discontinuity designs for cases where the outcome variable is itself discrete. When the assignment variable is discrete, standard confidence intervals do not have the nominal level of coverage, but confidence intervals with the correct coverage rate can be constructed conditional on an assumed upper bound for the second derivative of the expectation of the outcome conditional on the assignment variable. We propose a novel method for estimating this bound on the second derivative in cases where the outcome variable is generated by a binary outcome threshold-crossing model. The method leverages prior results that show that key primitives that determine the second derivative are semiparametrically identified conditional on mild shape restrictions motivated by theory and a known distribution of unobserved heterogeneity. Applying our method to examine the effect of the Social Security claiming age eligibility threshold of 62 on claims in the United States, we find that the size of the spike in claims at the age of eligibility is sensitive to assumptions regarding the distribution of unobserved reservation utilities.