  We study the estimation of treatment effects of a binary policy in
environments with a staggered treatment rollout. We propose a new estimator --
Sequential Synthetic Difference in Difference (Sequential SDiD) -- and
establish its theoretical properties in a linear model with interactive fixed
effects. Our estimator is based on sequentially applying the original SDiD
estimator proposed in Arkhangelsky et al. (2021) to appropriately aggregated
data. To establish the theoretical properties of our method, we compare it to
an infeasible OLS estimator based on the knowledge of the subspaces spanned by
the interactive fixed effects. We show that this OLS estimator has a sequential
representation and use this result to show that it is asymptotically equivalent
to the Sequential SDiD estimator. This result implies the asymptotic normality
of our estimator along with corresponding efficiency guarantees. The method
developed in this paper presents a natural alternative to the conventional DiD
strategies in staggered adoption designs.
