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

In this paper, we study difference-in-differences identification and estimation strategies when the parallel trends assumption holds after conditioning on covariates. We consider empirically relevant settings where the covariates can be time-varying, time-invariant, or both. We uncover a number of weaknesses of commonly used two-way fixed effects (TWFE) regressions in this context, even in applications with only two time periods. In addition to some weaknesses due to estimating linear regression models that are similar to cases with cross-sectional data, we also point out a collection of additional issues that we refer to as \textit{hidden linearity bias} that arise because the transformations used to eliminate the unit fixed effect also transform the covariates (e.g., taking first differences can result in the estimating equation only including the change in covariates over time, not their level, and also drop time-invariant covariates altogether). We provide simple diagnostics for assessing how susceptible a TWFE regression is to hidden linearity bias based on reformulating the TWFE regression as a weighting estimator. Finally, we propose simple alternative estimation strategies that can circumvent these issues.