Assume a difference-in-difference setup where the control and treatment groups are heterogeneous (different observable characteristics) but
The parallel trend assumption is verified in the pre-treatment period
The composition of the two groups does not change over time (i.e. the heterogeneity stays the same)
Is it true to say that in this case, the difference-in-difference estimate is unbiased?
My concern is that one could say that unobserved time-varying effects may have different impacts on the control and treatment groups in the post-treatment period given the heterogeneity of the two groups (even though we observe the parallel trend assumption in the pre-treatment period)