Regression to the mean in regression discontinuity design: bias and sensitivity analysis

Abstract When making causal inferences from observational data, researchers must consider the effects of confounding. In a regression discontinuity design (RDD), individuals receive a treatment based on whether they score below or above a threshold value measured on a continuous variable. By assuming continuous regression lines for the potential outcomes at the threshold, RDD methods remove the confounding bias in estimating the treatment effect at the threshold. This effect is estimated by the jump in the regression line for the observed outcome at the threshold. Although RDD methods have received deserved attention in economics, the social sciences, and epidemiology, we show that inferences from RDDs using local and global linear regression estimators are prone to regression to the mean bias in certain situations. A common situation where the bias occurs is when a running variable has a normal distribution and the cutoff is relatively far from the mean of this distribution. We derive the expression for the limiting bias in this case. In general, the bias occurs when some units receive (or do not receive) treatment when their running variable values are extreme relative to the typical value of the running variable. Through simulations, we show that the regression to the mean bias can lead to inflated type I error rates and bias toward the null in typical settings. Simulations show that the RTM effect can be different for different estimators. We develop a novel method to correct this bias and provide valid inferences. We verify our correction method in simulations and apply it to a real-life example of the incumbency advantage in U.S. House elections.