A key assumption in regression discontinuity analysis is that units cannot affect the value of their running variable through strategic behavior, or manipulation, in a way that leads to sorting on unobservable characteristics around the cutoff. Standard identification arguments break down if this condition is violated. This paper shows that treatment effects remain partially identified under weak assumptions on individuals' behavior in this case. We derive sharp bounds on causal parameters for both sharp and fuzzy designs, and show how additional structure can be used to further narrow the bounds. We use our methods to study the disincentive effect of unemployment insurance on (formal) reemployment in Brazil, where we find evidence of manipulation at an eligibility cutoff. Our bounds remain informative, despite the fact that manipulation has a sizable effect on our estimates of causal parameters.
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