1. What is Assumption 6 Conditional Parallel Trends?
Assumption 6 Conditional Parallel Trends assumes parallel trends hold across all time periods and treatment groups. It is similar to the general parallel trends assumption in Borusyak et al. (2021) and Roth et al. (2022). Other definitions weaken this assumption by restricting to never-treated groups or single pre-treatment periods. In multi-strata settings, it only needs to hold for strata with treated units. This assumption allows for straightforward identification of conditional group-time ATTs, with each stratum containing treated units acting as its own staggered difference-in-difference.
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2. How can researchers identify treatment effects when Assumption 6 does not hold?
When Assumption 6 does not hold, researchers can identify treatment effects by making an alternative identifying assumption and using a triple-differences design. In this setting, researchers assume that while conditional parallel trends may be violated, the violation is constant across strata. This approach allows for the identification of treatment effects even when the parallel trends assumption is not fully satisfied. The triple-differences design involves comparing the treatment effect across three different groups or strata, which helps to control for potential confounding factors and provides a more robust estimation of the treatment effect. By carefully selecting the strata and making appropriate assumptions, researchers can still draw meaningful conclusions about the impact of the treatment, even in situations where the parallel trends assumption is not met.
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3. What does Assumption 7 Constant violation entail?
Assumption 7 Constant violation entails that E[Y srt () - Y srt ' ()|G sr = g] - E[Y srt () - Y srt ' ()|G sr = g ' ] equals E[Y sr ' t () - Y sr ' t ' ()|G sr = g] - E[Y sr ' t () - Y sr ' t ' ()|G sr = g ' ] for all r = r ' , t = t ' , g = g '. This assumption generalizes the identification assumptions from Olden and Moen (2020) to settings with multiple strata where the treatment can be arbitrarily staggered in different strata. It restricts the control potential outcomes in a stratum r ' conditional on treatment assigned in stratum r, irrespective of the treatment distribution in r '. This assumption allows for parallel trends within any strata that receive treatment but may fail in a 'pure placebo' stratum where no unit receives treatment. Under Assumption 7, the conditional group-time ATT in stratum r can be identified by appending a second difference-in-differences to the result from Proposition 2. The identification result conditions on both the observed treatment in r and r ' in selecting valid units for the 'placebo' difference-in-differences. All observations used in the second difference-in-differences are under control, even if they may initiate treatment in the future. Researchers can aggregate the conditional group-time ATTs using researcher-specified weights into a single treatment effect summary, such as the weighted average across non-zero stratum-specific group-time ATTs.
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4. How can triple-differences approach be applied in pre-treatment periods?
The triple-differences approach can be applied in pre-treatment periods by incorporating additional pre-treatment periods in a conventional differences-in-differences setting, as demonstrated by Eigami and Yamauchi (2023). This approach allows for a more comprehensive analysis of the entire treatment history vector d = {d1, d2, ..., dT}. By considering separate, state-specific difference-in-differences designs that leverage treatment variation across industry and time, researchers can gain insights into the effects of staggered adoption designs. The potential outcomes can be defined in terms of the entire treatment history, summarized by the initiation time. This approach is particularly relevant in clusterrandomized experiments, where the choice between analyzing individual or cluster-averaged analyses is crucial. The cohort average treatment effect on the treated (CATT) is another measure that can be defined using the same quantity. Recent work by Esteve-Perez et al. (2020) provides evidence for the original null under an alternative estimation strategy. In cases with multiple placebo strata and differentially staggered treatment, the set of valid placebo comparisons will depend on the specific stratum-specific group-time ATT.
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