Nonparametric Bayes analysis of the sharp and fuzzy regression discontinuity designs

We develop an analysis of sharp and fuzzy RD designs, based on a new approach for non-parametrically estimating the unknown functions of the forcing variable and, in the case of the fuzzy design, a new modeling of the unobserved confounder. Several novel ideas are employed. First, in estimating the functions of the forcing variable, we include a knot at the threshold, which is not in general an observed value of the forcing variable, to al-low for curvature in the estimated functions from the breakpoint to the nearest values on either side of the breakpoint. Second, we cluster knots close to the threshold with the aim of controlling the approximation bias. Third, we introduce a new second-difference prior on the spline coefficients that can deal with unequally spaced knots. The number of knots and other features of the model are compared through marginal likelihoods and Bayes factors. Fourth, in the fuzzy case, instead of assuming that the confounder is continuous, as in the models to date, we suppose that the confounder is a discrete random variable that represents one of three exogenous subject-types, compliers, never-takers and always-takers. We show that under certain mild conditions, the effect of the forcing variable on the mean outcome