Event study controlling for unit-specific linear time trends I am running an event-study analysis. Is it possible to control for individual FE, year FE, and  individual-specific linear time trends, and still obtain coefficients for every year?
For example, I'd like to run:
$$ Y_{it} = \beta_0 + \sum_{j\neq 0}\beta_{1t} Treat_i\times Year_{t=j} + \beta_{2i} Unit_i + \beta_{3t} Year_t + \beta_{4i} (Unit_i \times t) + \varepsilon_{it}$$
Does this specification make sense? I know that it is possible to run a difference-in-difference specification (i.e., replacing the yearly coefficients with a coefficient for the entire post-period).
 A: 
Is it possible to control for individual FE, year FE, and individual-specific linear time trends, and still obtain coefficients for every year?

Yes. However, your model has redundancies. For instance, you interact $Treat_i$ with pre-period year dummies. Software will automatically attempt estimation of the constituent terms of your interaction, so coefficients will be estimated for $Treat_i$ and each pre-period year dummy. Note, your model also includes year fixed effects, which results in the estimation of $T-1$ dummies for years. Since your model already contains year dummies for all $j$ periods relative to the onset of treatment, some of the year effects must be omitted. In sum, you will still obtain coefficients on all of your year effects. Just be mindful of this should software spit out warning messages in regard to singularities.

Does this specification make sense?

Yes. Researchers often use event study estimates to buttress claims of trend equivalence before policy (treatment) adoption. You could also incorporate individual-specific linear time trends in event study settings, but it seems redundant in my opinion. Multiplying each individual-specific effect with a continuous linear time index is something I suggest you perform later in your analysis as a robustness check. Usually we are interested in whether treatment effects hold after we incorporate individual-specific linear time trends. This is typically performed in settings where we have acquired many pretreatment observations.

I know that it is possible to run a difference-in-difference specification (i.e., replacing the yearly coefficients with a coefficient for the entire post-period).

Yes. You indicated earlier that the timing of treatment is standardized; it begins at the same time for all units. To obtain your treatment effect, simply interact $Treat_{i}$ with one post-treatment indicator which indexes the entire post-treatment epoch. Note, this is different from your previous model where you interact treatment with individual period dummies in the years before treatment. In a typical event study, you are testing for differences between your treatment and control groups prior to the onset of treatment. Coefficients should be bounded around zero in the pre-period. In other words, you shouldn't be observing treatment effects before the actual treatment commences.
If you want to estimate a dynamic model for the pre- and post-exposure periods, you could also interact your treatment dummy with post-exposure period dummies as well. Reproducing equation 6 from this working paper, you could do the following:
$$
y_{i,t} = \alpha_{i} + \lambda_{t} + \sum_{\iota = -K}^{-2} \mu_{\iota} D^{\iota}_{i,t} + \sum_{\iota = 0}^{L} \mu_{\iota} D^{\iota}_{i,t} + \nu_{i,t},
$$
where $D^{\iota}_{i,t}$ is your interaction term from before, which we estimate for all K periods before treatment, and for all L periods after treatment. Again, we obtain $D^{\iota}_{i,t}$ via the interaction of treatment with individual year dummies.
Period $-1$ is the baseline year, or the period immediately before treatment adoption. Note, one period (year) must be excluded to avoid collinearity. You could alter the notation to suit your needs, just make sure you omit one year dummy. Researchers often omit the period before treatment, but I have seen papers where a more distant period relative to treatment was omitted. Try out these different methods and see what works in your setting.
