  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.
