Distributional Difference-in-Differences: Aggregation Before or After Quantile Inversion?

Staggered distributional difference-in-differences produces cohort-specific potential-outcome distributions, but applied work typically wants one overall quantile treatment effect. Two natural summaries---averaging cohort quantile treatment effects (QTTs) and mixing cohort distributions before inversion---use the same policy weights yet answer different target-population questions and can disagree even in sign. We derive the exact sharp interval for their gap conditional on cohort quantiles and weights, a globally sharp range-only envelope, and a locally sharp density-tilt representation. We develop joint smooth and mass-point-safe inference and show that neither estimator is uniformly more precise, even when the estimands coincide. In a same-object reconstruction of a public staggered-QTT application, holding data, identification, distributions, and weights fixed while changing only aggregation order reverses reported signs at several quantiles. Aggregation order is therefore part of the estimand and must be chosen before inversion.