What Do We Get from Two-Way Fixed Effects Regressions? Implications from Numerical Equivalence

Abstract This paper develops numerical and causal interpretations of two-way fixed effects (TWFE) regressions in settings with nonbinary, nonstaggered treatments and time-varying covariates. Using the equivalence between TWFE and pooled first-difference (FD) regressions, I express the TWFE coefficient as a weighted average of FD coefficients across all horizons, clarifying how short- and long-run changes contribute to the estimate. Causal interpretation of the TWFE coefficient relies on common trends assumptions at all horizons simultaneously, whereas each FD coefficient relies on the assumption only at its own horizon. This structure opens the identifying assumptions to empirical scrutiny: I propose diagnostic procedures that assess common trends horizon by horizon, and illustrate them by reexamining TWFE estimates of minimum-wage effects on employment.