I have two questions related to having fixed effects in the DD model.
I have a treatment that occurs at different times (e.g., 2001, 2005, etc.). I want to fit a DD model, so I standardize the treatment years to year "0" as the the treatment time. To control for treatment year heterogeneity, I included the true year fixed effects.
$y_{it} = \beta_0 + \beta_1 \text{Treat} + \beta_2 \text{After} + \beta_3 (\text{Treat $\cdot$ After}) + \eta (\text{Year Fixed Effects})+ \gamma C_{it} + \epsilon_{it}$
Question 1: Is there anything wrong with this model?
Question 2: Is there an issue with including time-constant fixed effects to this DD model? For example, what if I include i-level fixed effects ($\alpha_i$) and/or group indicators of i fixed effects (e.g. male/female or race)? I realize that DD cancels out time-constant i-lvl FE, but what if I include it here again?