Two-way fixed effects, the two-way mundlak regression, and difference-in-differences estimators

Abstract I derive a result on the equivalence between the two-way fixed effects (TWFE) estimator and an estimator obtained from a pooled ordinary least squares regression that includes unit-specific time averages and time-period-specific cross-sectional averages—the two-way Mundlak (TWM) regression. The equivalence between TWFE and TWM implies that various estimators used for intervention analysis can be computed using pooled OLS that controls for time-constant treatment cohort indicators, time-period indicators, covariates, and interactions among them—allowing for considerable treatment effect heterogeneity. An extended version of TWFE (ETWFE) is equivalent to the POLS approach. I show that an imputation estimator, derived under no anticipation and parallel trends assumptions, is also equivalent to the POLS/ETWFE estimator. The equivalence among various estimators shows that average treatment effects on the treated are identified by flexible regression. The framework allows for event study estimators, which can be used to test for pre-trends, and flexible estimation that allows for cohort-specific trends.