  Quantifying the impact of regulatory policies on social welfare generally
requires the identification of counterfactual distributions. Many of these
policies (e.g. minimum wages or minimum working time) generate mass points
and/or discontinuities in the outcome distribution. Existing approaches in the
difference-in-difference literature cannot accommodate these discontinuities
while accounting for selection on unobservables and non-stationary outcome
distributions. We provide a unifying partial identification result that can
account for these features. Our main identifying assumption is the stability of
the dependence (copula) between the distribution of the untreated potential
outcome and group membership (treatment assignment) across time. Exploiting
this copula stability assumption allows us to provide an identification result
that is invariant to monotonic transformations. We provide sharp bounds on the
counterfactual distribution of the treatment group suitable for any outcome,
whether discrete, continuous, or mixed. Our bounds collapse to the
point-identification result in Athey and Imbens (2006) for continuous outcomes
with strictly increasing distribution functions. We illustrate our approach and
the informativeness of our bounds by analyzing the impact of an increase in the
legal minimum wage using data from a recent minimum wage study (Cengiz, Dube,
Lindner, and Zipperer, 2019).
