Regression Discontinuity Designs Using Covariates∗

We study identification, estimation, and inference in Regression Discontinuity (RD) designs when additional covariates are included in the estimation. Standard RD estimation and inference is based on local polynomial regression methods using two variables: the outcome and the running variable that determines treatment assignment. Applied researchers often include additional covariates in their specifications to increase efficiency. However, no results justifying covariate adjustment have been formally derived in the RD literature, leaving applied researchers with little practical guidance and leading to a proliferation of ad-hoc methods that may result in invalid estimation and inference. We examine the properties of a local polynomial estimator that incorporates discrete and continuous covariates in an additive separable, linear-in-parameters way and imposes a common covariate effect on both sides of the cutoff. Under intuitive, minimal assumptions, we show that this covariate-adjusted RD estimator remains consistent for the standard RD treatment effect and characterize precisely the potential point estimation and inference improvements. In contrast, we show that estimating a specification with interactions between treatment status and the covariates leads to an estimator that is inconsistent in general. A key feature of all our results is that we do not impose any model assumptions, such as functional form restrictions or additional smoothness conditions, on the conditional expectation of the outcome given the running variable and additional covariates. We also present new asymptotic mean squared error expansions, optimal bandwidth choices, optimal point estimators, robust nonparametric inference procedures based on bias-correction techniques, and heteroskedasticity-consistent standard errors. Our results cover sharp, fuzzy, and kink RD designs, and we discuss extensions to clustered data. Finally, we include two empirical illustrations where we find 5% to 10% reduction in confidence interval length, and an extensive simulation study. All methods are implemented in companion R and Stata software packages.