Compositional difference-in-differences

Abstract

Many causal questions involve outcomes distributed across mutually exclusive categories, such as votes by party, employment by status, or electricity generation by energy source, where researchers care about both the shares and the total quantity. It is well known that linear Difference-in-Differences (DiD) methods applied to category shares can produce incoherent counterfactuals and lack a discrete-choice foundation. This paper develops Compositional Difference-in-Differences (CoDiD), a framework for causal inference with categorical and compositional outcomes. For changes in category shares, I represent the underlying outcomes through a discrete copula and introduce a discrete copula stability assumption: absent treatment, the dependence structure between outcomes and group membership would remain stable over time. This assumption has two equivalent interpretations: parallel changes in relative utilities in a random-utility model and parallel trajectories in the Aitchison geometry of the simplex. A stronger assumption, parallel growth in log-counts, jointly identifies treatment effects on both category shares and the total quantity. I also provide sharp bounds when the identifying assumption is relaxed and discuss principled strategies for handling zero cells. Applying CoDiD to early voting in the 2008 U.S. presidential election, I find that the policy increased turnout by 4.4\% and the Democratic vote share by 0.92 percentage points.