Extrapolating Away from the Cutoff in Regression Discontinuity Designs

Canonical RD designs yield credible local estimates of the treatment effect at the cutoff under mild continuity assumptions, but they fail to identify treatment effects away from the cutoff without additional assumptions. The fundamental challenge of identifying treatment effects away from the cutoff is that the counterfactual outcome under the alternative treatment status is never observed. This paper aims to provide a methodological blueprint to identify treatment effects away from the cutoff in various empirical settings by offering a non-exhaustive list of assumptions on the counterfactual outcome. Instead of assuming the exact evolution of the counterfactual outcome, this paper bounds its variation using the data and sensitivity parameters. The proposed assumptions are weaker than those introduced previously in the literature, resulting in partially identified treatment effects that are less susceptible to assumption violations. This approach accommodates both single cutoff and multi-cutoff designs. The specific choice of the extrapolation assumption depends on the institutional background of each empirical application. Additionally, researchers are recommended to conduct sensitivity analysis on the chosen parameter and assess resulting shifts in conclusions. The paper compares the proposed identification results with results using previous methods via an empirical application and simulated data. It demonstrates that set identification yields a more credible conclusion about the sign of the treatment effect.