Regression-based estimators in difference-in-differences with time-varying covariates

This study proposes a regression-based estimation method in difference-in-differences settings in the presence of time-varying covariates – a scenario commonly encountered in applications. We impose only a conditional parallel trends assumption with time-varying covariates and plausible assumptions on the conditional expectation functions. We show that a family of causal effect parameters are exactly the coefficient estimators from our proposed regressions even in the presence of staggered treatment timing and treatment effect heterogeneity across cohorts, time periods, and covariates. These parameters can be further aggregated to the dynamic treatment effects and the overall effect of being treated. We establish the corresponding asymptotic properties. Simulation studies suggest that our proposed regression-based estimators successfully outperform in estimating the causal parameters. Finally, we apply this method to evaluate the effect of intrastate bank deregulation on income inequality in the United States in the setting of Beck et al. (2010). We find substantially different results based on our proposed method.