Difference-in-Differences when Parallel Trends Holds Conditional on Covariates

We consider difference-in-differences identification and estimation strategies when the parallel trends assumption holds conditional on covariates, which can be time-varying, time-invariant, or both. We uncover several weaknesses of two-way fixed effects (TWFE) regressions in this context. The most important, which we call \textit{hidden linearity bias}, arises because transformations that eliminate unit fixed effects also transform the covariates, either implicitly changing the identification strategy or relying on correct model specification. We provide diagnostics for assessing a TWFE regression's susceptibility to hidden linearity bias and propose alternative estimation strategies that circumvent these issues.