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

TL;DR

This paper clarifies the numerical and causal interpretation of two-way fixed effects regressions with complex treatments, proposing diagnostics and illustrating their application to minimum-wage effects on employment.

Contribution

It establishes the equivalence between TWFE and pooled first-difference regressions, providing new insights into their interpretation with nonbinary, nonstaggered treatments.

Findings

TWFE coefficients can be expressed as weighted averages of first-difference coefficients.

Causal interpretation requires common-trends assumptions across all horizons.

Diagnostic procedures are proposed and applied to minimum-wage effects on employment.

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.