These are follow-up questions from this post. Reproducing the OP's equation below:
$$
y_{it} = \beta_{0} + \beta_{1}\text{Treat}_{i} + \sum_{j \neq k} \lambda_{j} \text{Year}_{t=j} + \sum_{j \neq k} \delta_j \left( \text{Treat}_i \times \text{Year}_{t=j} \right) + X_{it}'\gamma + \epsilon_{it},
$$
then each $\delta_{j}$ is a separate estimate of the treatment effect for each individual treatment year. As per your post, I assume you consider all $j \leq k$ as your pretreatment epoch. Thus, each coefficient on $\delta_{j}$ is an estimate of the $j$-th additive yearly treatment effect. Due to concerns in the comments, you should be more explicit about what periods you are indexing.
Adding multiple pre- and post-treatment periods is not difficult in this setting. Assume all treated units/entities receive treatment at the same time. Then, let $T_{0} + 1$ be the first time period at which treated units receive the treatment; this does not vary across units. I will also define $D_{it}$ as equal to unity for only those periods when treated units enter into the treatment condition. Note, this is simply the product term in the formulation above. Thus, $\text{Treat}_i \times \text{Year}_{t} = D_{it}$. For the control group, we have the following
$$
D^{C} = \begin{Bmatrix}
D_{it} = 0 \hspace{3pt} \forall \hspace{3pt} t
\end{Bmatrix}
$$
which is the baseline history of never receiving the treatment. For the treatment group, the binary treatment indicator is expressed as following:
$$
D^{T} = \begin{cases}
D_{it} = 0 \hspace{3pt} \forall \hspace{3pt} t \leq T_{0} \\
D_{it} = 1 \hspace{3pt} \forall \hspace{3pt} t > T_{0}
\end{cases}
$$
The periods $t \leq T_{0}$ are all the $k$ periods before treatment. The periods $t > T_{0}$ are all the $t$ periods during treatment (i.e., the post-treatment period). Thus, $\delta_{j}$ when $t = j$ is a unique estimate of the treatment effect in that year. I assume you are interested, separately, in each one of those effects.
To clear up any confusion regarding notation, you could also specify your equation more explicitly by labeling the limits of summation. Suppose you have yearly data on countries $i$ from 1980 to 2020. Now, say some population-level health intervention affects some countries in 2010, but not others. The intervention is effective beginning in 2010 and remains in place indefinitely. If you want to investigate effects in each individual exposure year, then the classical difference-in-differences equation generalizes to the following:
$$
y_{it} = \beta_{0} + \beta_{1}\text{Treat}_{i} + \sum_{j = 2010}^{2020} \lambda_{j} \text{Year}_{t=j} + \sum_{j = 2010}^{2020} \delta_j \left( \text{Treat}_i \times \text{Year}_{t=j} \right) + X_{it}'\gamma + \epsilon_{it},
$$
where the treatment dummy is interacted—separately—with a post-intervention year (time) dummy. This results in a separate main effect for each post-treatment year, and a separate estimate of the treatment effect for each exposure year. All $t$ periods before 2010 serve as the pre-treatment epoch, and remain coded 0 to reflect that reality.
What is the meaning of the coefficient $\beta_{0}$ and $\beta_{1}$?
I know that the meaning in regular pre-post regression but not sure what is the meaning in this case.
This is still an interaction model, even with multiple pre- and post-treatment periods. We interpret the estimate of $\beta_{0}$ as the mean of your outcome for the control group in the years before treatment is adopted. Your estimate of $\beta_{1}$ is the expected mean difference in your outcome between treatment and control groups in the pre-treatment period. In your case, this is when all $\text{Year}_{t}$ periods are equal to 0 (i.e., all $k$ periods before treatment). This can viewed as the "baseline difference" in your outcome between the two groups. The estimated coefficient(s) associated with your interaction term(s) should be your focus.
Just to be sure, there is no way I can sum what is the total influence of the treatment over Y? I can just tell what is the treatment effect in a specific year? I'm trying to understand how can I analyze the results I got from the regression.
In this case, interact the treatment variable with one post-treatment indicator indexing all post-treatment periods, irrespective of the unit's group status. In the foregoing equation, you are investigating treatment effects in each year individually. It would help if you posted your output as we are only working with abstract equations.
Best Answer
Both methods are doing something very similar: comparing trends between treatment and control. The first has the advantage of being able to easily test the joint null that all the pre $\delta$s are zero, rather than just comparing the coefficients visually, one by one. Moreover, the pooled model will probably yield more precise estimates than two separate models.
The second might be easier to explain to folks that don't understand interactions and makes it easier to plot the coefficients (rather than their difference).