  This paper considers identification and estimation of causal effect
parameters from participating in a binary treatment in a difference in
differences (DID) setup when the parallel trends assumption holds after
conditioning on observed covariates. Relative to existing work in the
econometrics literature, we consider the case where the value of covariates can
change over time and, potentially, where participating in the treatment can
affect the covariates themselves. We propose new empirical strategies in both
cases. We also consider two-way fixed effects (TWFE) regressions that include
time-varying regressors, which is the most common way that DID identification
strategies are implemented under conditional parallel trends. We show that,
even in the case with only two time periods, these TWFE regressions are not
generally robust to (i) time-varying covariates being affected by the
treatment, (ii) treatment effects and/or paths of untreated potential outcomes
depending on the level of time-varying covariates in addition to only the
change in the covariates over time, (iii) treatment effects and/or paths of
untreated potential outcomes depending on time-invariant covariates, (iv)
treatment effect heterogeneity with respect to observed covariates, and (v)
violations of strong functional form assumptions, both for outcomes over time
and the propensity score, that are unlikely to be plausible in most DID
applications. Thus, TWFE regressions can deliver misleading estimates of causal
effect parameters in a number of empirically relevant cases. We propose both
doubly robust estimands and regression adjustment/imputation strategies that
are robust to these issues while not being substantially more challenging to
implement.
