I have questions about the case when I employ fixed effect in Difference-in-Difference(DID). As I understand, both treatment group dummy(D_i) and treatment time dummy(D_t) should be independent from error term in regression. My questions are as follows:
If we consider fixed effect in DID, is it against the exogeneity assumption for DID? It is because there is a correlation between group dummy and fixed effect.
I also know that when we put fixed effect in DID, statistical program, for example Stata, is automatically dropping out time-invariant group dummy. If so, do we still interpret interaction terms between group dummy(D_i) and time dummy(D_t) as average treatment on the treated(ATT)? My understanding is that although time-invariant variable is automatically dropped, the interaction term(D_i*D_t) still has ATT meaning. This is because fixed effect has identical information with treatment group dummy(D_i) that is dropped. Am I correct? It looks like, but I am not confident in a matrix format. If it is different, could you guide me using some math?
If my understanding, treatment group dummy D_i has identical information with fixed effect because D_i is automatically dropped, is correct, what is the difference between DID model without fixed effect and DID with fixed effect?
What is benefit/cost from employing fixed effect to the matched-sample given that parallel assumption holds? Can we argue that including fixed effect in DID is always better than DID without fixed effect? If not, when should we put fixed effect and when we shouldn't in DID model?
Thank you for your help in advance.