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In any multiperiod panel, a two-way fixed effects (TWFE) regression isnumerically equivalent to a first-difference (FD) regression that pools allpossible between-period gaps. Building on this observation, this paper developsnumerical and causal interpretations of the TWFE coefficient. At the samplelevel, the TWFE coefficient is a weighted average of FD coefficients withdifferent between-period gaps. This decomposition is useful for assessing thesource of identifying variation for the TWFE coefficient. At the populationlevel, a causal interpretation of the TWFE coefficient requires a common trendsassumption for any between-period gap, and the assumption has to be conditionalon changes in time-varying covariates. I propose a natural generalization ofthe TWFE estimator that can relax these requirements.

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