The method of difference-in-differences (DID) is widely used to study the causal effect of policy interventions in observational studies. DID exploits a before and after comparison of the treated and control units to remove the bias due to time-invariant unmeasured confounders under the parallel trends assumption. Estimates from DID, however, will be biased if the outcomes for the treated and control units evolve differently in the absence of treatment, namely if the parallel trends assumption is violated. We propose a general identification strategy that leverages two groups of control units whose outcomes relative to the treated units exhibit a negative correlation, and achieves partial identification of the average treatment effect for the treated. The identified set is of a union bounds form that previously developed partial identification inference methods do not apply to. We develop a novel bootstrap method to construct  valid confidence intervals for the identified set and parameter of interest when the identified set is of a union bounds form, and we establish the  theoretical properties. We develop a simple falsification test and sensitivity analysis. We apply the proposed strategy for bracketing to an application on the effect of voter identification laws in URLia and Indiana on turnout and find evidence that the laws increased turnout rates .
The method of difference-in-differences (DID) is widely used to study the causal effect of policy interventions in observational studies. DID employs a before and after comparison of the treated and control units to remove bias due to time-invariant unmeasured confounders under the parallel trends assumption. Estimates from DID, however, will be biased if the outcomes for the treated and control units evolve differently in the absence of treatment, namely if the parallel trends assumption is violated. We propose a general identification strategy that leverages two groups of control units whose outcomes relative to the treated units exhibit a negative correlation, and achieves partial identification of the average treatment effect for the treated. The identified set is of a union bounds form that involves the minimum and maximum operators, which makes the canonical bootstrap generally inconsistent and naive methods overly conservative. By utilizing the directional inconsistency of the bootstrap distribution, we develop a novel bootstrap method to construct uniformly valid confidence intervals for the identified set and parameter of interest when the identified set is of a union bounds form, and we establish the method's theoretical properties. We develop a simple falsification test and sensitivity analysis. We apply the proposed strategy for bracketing to study whether minimum wage laws affect employment levels .
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exploits
employs
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the
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previously developed partial identification inference methods do not apply to. We
involves the minimum and maximum operators, which makes the canonical bootstrap generally inconsistent and naive methods overly conservative. By utilizing the directional inconsistency of the bootstrap distribution, we
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uniformly
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method's
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an application on the effect of voter identification laws in URLia and Indiana on turnout and find evidence that the laws increased turnout rates
study whether minimum wage laws affect employment levels