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

In any multiperiod panel, a two-way fixed effects (TWFE) regression is numerically equivalent to a first-difference (FD) regression that pools all possible between-period gaps. Building on this observation, this paper develops numerical and causal interpretations of the TWFE coefficient. At the sample level, the TWFE coefficient is a weighted average of FD coefficients with different between-period gaps. This decomposition improves transparency by revealing the sources of variation that the TWFE coefficient captures. At the population level, causal interpretation of the TWFE coefficient relies on a common trends assumption for any between-period gap, conditional on changes, not levels, of time-varying covariates. I propose a simple modification to the TWFE approach that naturally eases this assumption.

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