Regression discontinuity inference with specification error

A regression discontinuity (RD) research design is appropriate for program evaluation problems in which treatment status (or the probability of treatment) depends on whether an observed covariate exceeds a fixed threshold. In many applications the treatment-determining covariate is discrete. This makes it impossible to compare outcomes for observations "just above" and "just below" the treatment threshold, and requires the researcher to choose a functional form for the relationship between the treatment variable and the outcomes of interest. We propose a simple econometric procedure to account for uncertainty in the choice of functional form for RD designs with discrete support. In particular, we model deviations of the true regression function from a given approximating function-the specification errors-as random. Conventional standard errors ignore the group structure induced by specification errors and tend to overstate the precision of the estimated program impacts. The proposed inference procedure that allows for specification error also has a natural interpretation within a Bayesian framework.