🤖 AI Summary
Standard difference-in-differences (DID) methods struggle to identify counterfactual distributions under regulatory policies—such as minimum wage laws—when confronted with mass points, distributional discontinuities, nonstationarity, or unobserved selection bias. This paper proposes a unified partial identification framework grounded in a copula stability assumption, applicable to discrete, continuous, and mixed outcome variables. Under continuity and monotonicity, the framework collapses to the point-identification result of Athey & Imbens (2006), and it is transformation-invariant. Integrating DID, copula modeling, and partial identification theory, the approach yields sharp bounds on the counterfactual distribution. Empirically, it precisely quantifies the causal impact of minimum wage increases on the joint distribution of employment and earnings. The resulting bounds are highly informative, substantially extending both the applicability and robustness of policy evaluation methods in settings where conventional DID assumptions fail.
📝 Abstract
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).
