Quantile and Distribution Treatment Effects on the Treated with Possibly Non-Continuous Outcomes

Applied Difference-in-Differences studies often involve outcomes that are discrete, mixed, censored, or otherwise non-continuously distributed, while policy questions frequently concern distributional effects rather than mean effects alone. This paper develops a distributional DiD framework for identifying and conducting uniform inference on distribution and quantile treatment effects on the treated in such settings under stated identifying and regularity conditions. Identification is based on distributional parallel trends and no-anticipation assumptions, illustrated through an economic model of crime that generates count-valued untreated potential outcomes. The identification and asymptotic theory accommodate staggered treatment adoption and a general sampling scheme encompassing repeated cross-sections, unbalanced panels, rotating panels, and balanced panels. The paper also proposes a test of functional over-identifying restrictions as a diagnostic for the identifying assumptions and working-CDF specification. An empirical application to the effect of police on crime illustrates the practical relevance of the approach and shows how distributional effects can be interpreted as event-probability effects for count outcomes.