Behavior of estimators as the number of periods and cohorts grows

Ascertain how the proposed estimators for the counterfactual distribution and derived quantile treatment effects behave as the number of time periods T (and the number of treatment cohorts R) increases, particularly when per-cohort sample sizes become small.

Background

The paper evaluates finite-sample performance of the proposed nonparametric estimators via Monte Carlo experiments and notes theoretical expectations for good performance when the sample size is large relative to the number of periods and cohorts. However, the implications when the number of periods (and, consequently, cohorts) grows are not theoretically characterized.

This uncertainty is practically important because increasing T can fragment the sample across many cohorts, reducing per-cohort sample sizes and potentially affecting estimator bias, variance, and inference.

References

What happens when $T$ (and consequently $R$) grows is unclear, as the number of units in a given cohort can be very small.

— Distributional Difference-in-Differences Models with Multiple Time Periods  (2408.01208 - Ciaccio, 2024) in Section 4.1, Monte-Carlo Simulations — varying N and T

When the cells are estimated, Zhou et al.~\citep{zhou2025} give consistency and rates by empirical-process arguments, with distributional inference open by their own account; when the cells are themselves Fr\'echet means of samples, their asymptotic covariance is the sandwich eq:if-sandwich with the covariant Hessian of \autoref{subsubsec:rob-lcf}, the central limit theorem for intrinsic means being that of Bhattacharya and Patrangenaru \citep{bhattacharyapatrangenaru2005}, under their regularity assumptions; the Fr\'echet variance of a cell, the minimum of its objective, has the limit of Dubey and M\"uller \citep{dubeymueller2019}.

eq:if-sandwich:

IF(x)=−H−1 ψ(x,T(F)),Ω=CovF(ψ(X,T(F))),Avar=H−1 Ω H−⊤=∫IF IF dF.\begin{aligned} \mathrm{IF}(x)&= -H^{-1}\,\psi\big(x,T(F)\big), \qquad \Omega = Cov_F\big(\psi(X,T(F))\big),\\ \mathrm{Avar}&= H^{-1}\,\Omega\,H^{-\top} = \int \mathrm{IF}\,\mathrm{IF}\,dF. \end{aligned}

— Estimators from Convex Geometries of the Simplex  (2610.21551 - Navas, 15 Oct 2026) in Section 5.3, “Compositional difference-in-differences and the choice of transport”