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Quantifying the impact of regulatory policies on social welfare generallyrequires the identification of counterfactual distributions. Many of thesepolicies (e.g. minimum wages or minimum working time) generate mass pointsand/or discontinuities in the outcome distribution. Existing approaches in thedifference-in-difference literature cannot accommodate these discontinuitieswhile accounting for selection on unobservables and non-stationary outcomedistributions. We provide a unifying partial identification result that canaccount for these features. Our main identifying assumption is the stability ofthe dependence (copula) between the distribution of the untreated potentialoutcome and group membership (treatment assignment) across time. Exploitingthis copula stability assumption allows us to provide an identification resultthat is invariant to monotonic transformations. We provide sharp bounds on thecounterfactual distribution of the treatment group suitable for any outcome,whether discrete, continuous, or mixed. Our bounds collapse to thepoint-identification result in Athey and Imbens (2006) for continuous outcomeswith strictly increasing distribution functions. We illustrate our approach andthe informativeness of our bounds by analyzing the impact of an increase in thelegal minimum wage using data from a recent minimum wage study (Cengiz, Dube,Lindner, and Zipperer, 2019).

Evaluating the Impact of Regulatory Policies on Social Welfare in Difference-in-Difference Settings