Compositional difference-in-differences

Many causal questions concern vectors of quantities across mutually exclusive categories, votes by party, employment by status, generation by energy source, where both the shares and the total matter.

Standard practice runs a separate linear difference-in-differences (DiD) on each share, which violates the simplex constraint, cannot describe reallocation across categories, and is silent on the total.

This paper develops Compositional Difference-in-Differences (CoDiD).

For the share margin, parallel trends in log-odds identify the counterfactual composition in closed form, always inside the simplex, and admit two readings: parallel evolution of relative utilities in a random-utility model, and parallel trajectories in the Aitchison geometry of the simplex.

Strengthening the assumption to parallel growth in log-counts jointly identifies effects on shares and on the total, and reveals a composition adjustment factor, the ratio of inclusive-value growth across groups, that corrects the aggregation bias of log-total DiD.

I derive the joint asymptotic distribution of all treatment-effect estimators under multinomial sampling, provide a pre-trends test, and obtain sharp partial identification bounds when pre-treatment growth differentials are only bounded.

Applying CoDiD to early voting in the 2008 U.S.\ presidential election, I find a $4.4\%$ increase in turnout and a $0.92$ percentage-point increase in the Democratic vote share.