  We present a new identification condition for regression discontinuity
designs. We replace the local randomization of Lee (2008) with two restrictions
on its threat, namely, the manipulation of the running variable. Furthermore,
we provide the first auxiliary assumption of McCrary's (2008) diagnostic test
to detect manipulation. Based on our auxiliary assumption, we derive a novel
expression of moments that immediately implies the worst-case bounds of Gerard,
Rokkanen, and Rothe (2020) and an enhanced interpretation of their target
parameters. We highlight two issues: an overlooked source of identification
failure, and a missing auxiliary assumption to detect manipulation. In the case
studies, we illustrate our solution to these issues using institutional details
and economic theories.
