How Can Comparison Groups Strengthen Regression Discontinuity Designs?

In this paper, the authors examine some of the ways that different types of non-equivalent comparison groups can be used to strengthen causal inferences based on regression discontinuity design (RDD). First, they consider a design that incorporates pre-test data on assignment scores and outcomes that were collected either before the treatment became available or before the practice of assigning treatments based on a cut-off score began. The idea is to use these pre-test data to establish a baseline estimate of the relationship between the outcome variable and the assignment variable. Second, they evaluate a design that incorporates data on the assignment scores and outcomes of a single contemporaneous comparison group of units that are always ineligible for treatment. Here the idea is to establish baseline differences in the relationship between outcomes and assignment scores that prevail in the RD group and the comparison group. Third, they consider how unit level and group level covariates might be used to form an optimal control group from a pool of several candidate control groups. They explore how various methods of matching and reweighting can be used to construct a control group in which the functional relationship between the outcome and the assignment score closely resembles the relationship that prevails in the RD sample below the assignment cut-off value. In all three cases, they evaluate the statistical and behavioral assumptions that are required or the comparison group augmented RDD to produce unbiased estimates of specific treatment effects of interest. They also compare the assumptions required to extrapolate from the cut-off subpopulation to other sub-populations in the augmented RDD with the assumptions required to extrapolate from standard RDD.