Complex Discontinuity Designs Using Covariates
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Regression discontinuity designs are extensively used for causal inference in observational studies. However, they are usually confined to settings with simple treatment rules, determined by a single running variable, with a single cutoff. In this paper, we propose a new framework and methods for complex discontinuity designs that encompasses multiple treatment rules. These rules may be determined by multiple running variables, each with many cutoffs, that possibly lead to the same treatment. Moreover, the running variables may be discrete and the treatments do not need to be binary. In this framework, the observed covariates play a central role for identification, estimation, and generalization of causal effects. Identification is non-parametric and relies on a local strong ignorability assumption; that is, on local unconfoundedness and local positivity assumptions. Estimation proceeds as in any observational study under strong ignorability, yet in a neighborhood of the cutoffs of the running variables. We discuss estimation approaches based on matching and weighting, including additional regression adjustments in the spirit of doubly robust estimators. We present assumptions for generalization; that is, for identification and estimation of average treatment effects for target populations beyond the study sample that reside in a neighborhood of the cutoffs. We also propose two approaches to select the neighborhood for the analyses and assess the plausibility of the assumptions. We motivate and illustrate this framework with an example of the impact of grade retention on educational and juvenile crime outcomes.
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