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The regression discontinuity (RD) design is a popular quasi-experimental methods for applied causal inference. The method is sensitive to the assumption that individuals cannot precisely control their value of a "running variable" that determines treatment status. If subjects' scores can be precisely manipulated, then point identification is lost. We propose a procedure for obtaining partial identification bounds in the case of a discrete running variable under manipulation. Our method relies on two stages: first, we derive the distribution of non-manipulators under several assumptions. Second, we obtain bounds on the causal effect by solving a convex program. We propose methods for tightening the bounds using auxiliary covariates, and derive confidence intervals via the bootstrap.
We demonstrate the utility of our approach on a dataset of blood donations provided by Sheikh Khalifa Medical City and the Abu Dhabi Blood Bank. The data show that potential donors' reported hemoglobin levels are manipulated over a threshold to facilitate donation. Using our methods, we are able to obtain bounds on the treatment effect of an accepted donation on future donation behavior.
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