Estimation of Regression Discontinuity and Kink Designs with Multiple Running Variables

In regression discontinuity designs with multiple running variables (MRD), units are assigned different treatments based on whether their values on several observed running variables exceed known thresholds. In such designs, applied work commonly analyzes each running variable separately, estimating a single-dimensional RD design in the first running variable after limiting the sample to the set of individuals qualifying on the second threshold, and vice versa. In this paper, I propose a new estimator for MRD designs using thin plate splines that improves upon the applied practice in two ways. First, the estimator can be used to estimate the conditional average treatment effect at every point on the boundary separating treated and untreated units, and second, it provides efficiency gains by using the entire sample. I also develop analogous estimators for multidimensional regression kink (MRK) and multidimensional regression discontinuity/kink designs (MRDK). I show consistency of these estimators and construct asymptotically valid confidence intervals (CIs), before presenting simulation results showing that they produce estimates and CIs that perform well in finite samples. Finally, I demonstrate the performance of my MRD estimator with two empirical applications: Londoño-Vélez, Rodríguez, and Sánchez (2020) on the effect of financial aid on college enrollment, and Keele and Titiunik (2015) on the effect of political ads on election turnout. R code for estimation and inference will soon be available