  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.
