Two-Way Fixed Effects and Difference-in-Differences Estimators with Heterogeneous Treatment Effects and Imperfect Parallel Trends

Two-way fixed effects (TWFE) regressions with period and group fixed effects are widely used to estimate policies' effects: 26 of the 100 most cited papers published by the American Economic Review from 2015 to 2019 estimate such regressions. Researchers have long thought that TWFE estimators are equivalent to differences-in-differences (DID) estimators, that rely on a partly testable parallel trends assumption. In two-groups two-periods designs where a treatment group is untreated at both dates and a treatment group becomes treated at the second period, the treatment coefficient in a TWFE is indeed equivalent to a DID. Motivated by this fact, researchers have also estimated TWFE regressions in more complicated designs with many groups and periods, variation in treatment timing, treatments switching on and off, and/or non-binary treatments, confident that there as well, TWFE was giving them an estimation method that only relied on a partly testable parallel trends assumption. Two recent strands of literature have shattered that confidence. First, it has recently been shown that even if parallel trends holds, TWFE may produce misleading estimates, if the policy's effect is heterogeneous between groups or over time, as is often the case. The realization that one of the most commonly used empirical methods in the quantitative social sciences relies on an often-implausible assumption has spurred a flurry of methodological papers. Some of them have diagnosed this issue and analyzed its origins. Other papers have proposed alternative estimators relying on parallel trends conditions, like TWFE estimators, but robust to heterogeneous effects, unlike TWFE estimators. Hereafter, those alternative estimators are referred to as heterogeneity-robust DID estimators. Second, in a recent paper, Roth (2022) has shown that tests of the parallel trends assumption often lack statistical power, and may fail to detect differential trends between treated and control locations that are often large enough to account for a significant share of the policy's estimated effect. This realization has spurred a growing interest among practitioners for a second strand of literature, that has proposed alternative estimation methods relying on weaker assumptions than parallel trends. Examples include estimators relying on a conditional parallel trends assumption (see, e.g., Abadie, 2005), estimators assuming bounded differential trends (see, e.g., Manski and Pepper, 2018; Rambachan and Roth, 2023), estimators assuming a factor model with interactive fixed effects (see, e.g., Bai, 2003) and synthetic control estimators (see, e.g., Abadie et al., 2010), and estimators assuming grouped patterns of heterogeneity (see,e.g., Bonhomme and Manresa, 2015).This textbook aims to provide an overview of these two strands of literature, as well as other panel data methods routinely used for causal inference by practitionners.