Two-Way Fixed Effects, the Two-Way Mundlak Regression, and Difference-in-Differences Estimators

Abstract

I establish 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, which I call the two-way Mundlak (TWM) regression. This equivalence furthers our understanding of the anatomy of TWFE, and has several applications. The equivalence between TWFE and TWM implies that various estimators used for intervention analysis – with a common entry time into treatment or staggered entry, with or without covariates – can be computed using TWFE or pooled OLS regressions that control for time-constant treatment intensities, covariates, and interactions between them. The approach allows considerable heterogeneity in treatment effects across treatment intensity, calendar time, and covariates. The equivalence implies that standard strategies for heterogeneous trends are available to relax the common trends assumption. Further, the two-way Mundlak regression is easily adapted to nonlinear models such as exponential models and logit and probit models.