Compositional difference-in-differences for categorical outcomes

In difference-in-differences (DiD) settings with categorical outcomes, such as voting, occupation, or major choices, treatments often affect both total counts (e.g., turnout) and category shares (e.g., vote shares). Traditional linear DiD models can yield invalid counterfactuals in this context (e.g., negative values) and lack compatibility with standard discrete choice models. I propose Compositional DiD (CoDiD), which identifies counterfactual totals and shares under a parallel growths assumption: absent treatment, each category's size grows or shrinks at the same proportional rate in treated and control groups. I show that under a random utility model, this is equivalent to parallel trends in expected utilities, i.e., the change in average latent desirability for each alternative is identical across groups. Consequently, relative preferences (e.g., how individuals prefer Democrat vs. Republican) evolve in parallel, and counterfactual distributions follow parallel trajectories in the probability simplex. I extend CoDiD to (i) derive bounds under relaxed assumptions, (ii) accommodate staggered treatment timing, and (iii) construct a synthetic DiD analog. I illustrate the method's empirical relevance through two applications: first, I examine how early voting reforms affect voter choice in U.S. presidential elections; second, I analyze how the Regional Greenhouse Gas Initiative (RGGI) affected the electricity mix in participating states.