The moment is here: a generalised class of estimators for fuzzy regression discontinuity designs

The standard fuzzy regression discontinuity (FRD) estimator is a ratio of differences of local polynomial estimators. I show that this estimator does not have finite moments of any order in finite samples, regardless of the choice of kernel function, bandwidth, or order of polynomial. This leads to an imprecise estimator with a heavy-tailed sampling distribution, and inaccurate inference with small sample sizes or when the discontinuity in the probability of treatment assignment at the cutoff is small. I present a generalised class of computationally simple FRD estimators, which contains a continuum of estimators with finite moments of all orders in finite samples, and nests both the standard FRD and sharp (SRD) estimators. The class is indexed by a single tuning parameter, and I provide simple values that lead to substantial improvements in median bias, median absolute deviation and root mean squared error. These new estimators remain very stable in small samples, or when the discontinuity in the probability of treatment assignment at the cutoff is small. Simple confidence intervals that have strong coverage and length properties in small samples are also developed. The improvements are seen across a wide range of models and using common bandwidth selection algorithms in extensive Monte Carlo simulations. The improved stability and performance of the estimators and confidence intervals is also demonstrated using data on class size effects on educational attainment.