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A key assumption in regression discontinuity analysis is that units cannot manipulate the value of their running variable in a way that guarantees or avoids assignment to the treatment. Standard identification arguments break down if this condition is violated. This paper shows that treatment effects remain partially identified in this case. We derive sharp bounds on the treatment effects, show how to estimate them, and propose ways to construct valid confidence intervals. Our results apply to both sharp and fuzzy regression discontinuity designs. We illustrate our methods by studying the effect of unemployment insurance on unemployment duration in Brazil, where we find strong evidence of manipulation at eligibility cutoffs.

In the canonical regression discontinuity (RD) design for applicants who face an award or admissions cutoff, causal effects are nonparametrically identified for those near the cutoff. The effect of treatment on inframarginal applicants is also of interest, but identification of such effects requires stronger assumptions than those required for identification at the cutoff. This paper discusses RD identification away from the cutoff. Our identification strategy exploits the availability of dependent variable predictors other than the running variable. Conditional on these predictors, the running variable is assumed to be ignorable. This identification strategy is illustrated with data on applicants to Boston exam schools. Functional-form-based extrapolation generates unsatisfying results in this context, either noisy or not very robust. By contrast, identification based on RD-specific conditional independence assumptions produces reasonably precise and surprisingly robust estimates of the effects of exam school attendance on inframarginal applicants. These estimates suggest that the causal effects of exam school attendance for 9th grade applicants with running variable values well away from admissions cutoffs differ little from those for applicants with values that put them on the margin of acceptance. An extension to fuzzy designs is shown to identify causal effects for compliers away from the cutoff.