Compositional difference-in-differences

Abstract

Many causal questions involve outcomes distributed across mutually exclusive categories, such as votes by party or employment by status, where both category shares and the total quantity are of interest. This paper develops Compositional Difference-in-Differences (CoDiD), a framework for causal inference with categorical and compositional outcomes. Focusing first on shares, I introduce parallel trends in log-odds (PTLO), which requires the untreated log-odds of each category relative to a reference category to evolve in parallel across groups. I show that PTLO admits three equivalent interpretations: stability over time of the dependence structure between untreated categorical outcomes and group membership; parallel evolution of relative utilities in a random-utility model when shares arise from discrete choices; and parallel trajectories in the Aitchison geometry of the simplex. A stronger assumption, parallel growth in log-quantities, jointly identifies treatment effects on category shares and the total quantity. I also derive sharp bounds under a relaxation of the identifying assumption. Finally, I illustrate the framework by examining the effects of early voting on U.S. presidential elections.