A general bracketing strategy in Difference-in-Differences: Multi-Factor Models and Control Group Selection

Music City Center 

The difference-in-differences method estimates causal effects under unmeasured confounding, relying on the parallel trends assumption. Ye et al. (2023) relaxed this by introducing the bracketed trends assumption, requiring two control groups with negatively correlated outcome trends relative to the treatment group. While this enables partial identification of the average treatment effect on the treated, uncertainty accumulates over time, making the bounds overly conservative. Identifying feasible control groups also remains a challenge.
We address these issues by proposing an alternative bracketed trends assumption, requiring negative correlation over a period rather than at each step. We explore its relationship with the original assumption and establish sufficient conditions for the assumption to hold under a multi-factor interactive fixed effects model. Factor loadings guide the selection of control units, and we provide an asymptotically valid statistical inference procedure. We validate our approach through simulations and apply it to estimate the effect of the Election Day Registration policy on U.S. voter turnout.

Bayesian causal inference

missing not at random

noncompliance

Principal stratification

Rubin Causal Model

two-stage randomized design