  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 is useful for assessing the
source of identifying variation for the TWFE coefficient. At the population
level, a causal interpretation of the TWFE coefficient requires a common trends
assumption for any between-period gap, and the assumption has to be conditional
on changes in time-varying covariates. I propose a natural generalization of
the TWFE estimator that can relax these requirements.
