Group-specific linear trends and the triple-differences in time design

Differences-in-differences designs for estimating causal effects rely on an assumption of ``parallel trends" -- that in the absence of the intervention, treated units would have followed the same outcome trajectory as observed in control units. When parallel trends fails, researchers often turn to alternative strategies that relax this identifying assumption. One popular approach is the inclusion of a group-specific linear time trend in the commonly used two-way fixed effects (TWFE) estimator. In a setting with a single post-treatment and two pre-treatment periods it is well known that this is equivalent to a non-parametric ``triple-differences" estimator which is valid under a ``parallel trends-in-trends" assumption (Egami and Yamauchi, 2023). This paper analyzes the TWFE estimator with group-specific linear time trends in the more general setting with many pre- and post-treatment periods. It shows that this estimator can be interpreted as an average over triple-differences terms involving both pre-treatment and post-treatment observations. As a consequence, this estimator does not identify a convex average of post-treatment ATTs without additional effect homogeneity assumptions even when there is no staggering in treatment adoption. A straightforward solution is to make the TWFE specification fully dynamic with a separate parameter for each relative treatment time. However, identification requires that researchers omit at least two pre-treatment relative treatment time indicators to estimate a group-specific linear trend. The paper shows how to properly extend this estimator to the staggered adoption setting using the approach of Sun and Abraham (2021), correcting a perfect collinearity error in recent implementations of this method in Hassell and Holbein (2024). It concludes with a note of caution for researchers, showing through a replication of Kogan (2021) how inferences from group-specific time trend specifications can be extremely sensitive to arbitrary specification choices when parallel trends violations are present but do not follow an easily observed functional form.