Difference-in-differences via common correlated effects

We study the effect of treatment on an outcome when parallel trends hold conditional on an interactive fixed effects structure. In contrast to the majority of the literature, we propose identification using time-varying covariates. We assume the untreated outcomes and covariates follow a common correlated effects (CCE) model, where the covariates are linear in the same common time effects. We then demonstrate consistent estimation of the treatment effect coefficients by imputing the untreated potential outcomes in post-treatment time periods. Our method accounts for treatment affecting the distribution of the control variables and is valid when the number of pre-treatment time periods is small. We also decompose the overall treatment effect into estimable direct and mediated components.