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 regressions, I express the TWFE coefficient as a weighted average of first-difference coefficients across all horizons, clarifying how short- and long-run changes contribute to the estimate. Causal interpretation relies on common-trends assumptions across all horizons and conditioning on covariate changes rather than levels. I propose diagnostic procedures to assess these assumptions across horizons and illustrate them by reexamining TWFE estimates of minimum-wage effects on employment.