Regression discontinuity (RD) research designs identify the causal impact of a treatment using the idea that the rule governing the assignment of treatment to individuals is arbitrary. RD’s are typically used in scenarios where the treatment status of an individual (Di) is a deterministic and discontinuous function of an observable characteristic (Xi), e.g., Di =1 if Xi≥x0 and Di =0 if Xi<x0. In general, a simple regression of the form Yi = f(Xi) + βDi + ui formalizes the RD set-up, where β is the causal effect of interest. The key difference between this regression and a simple OLS (discussed here) is that Di is a deterministic function of Xi. The causal effect β is identified by distinguishing the discontinuous step-function, 1(Xi≥x0), from the smooth function, f(Xi). It should be noted that f(Xi) may or may not be linear in Xi, just like an OLS set-up need not be linear in the explanatory variables, but only in the parameters.
