Interpretation of event study difference-in-difference coefficient I understand mathematically what a difference-in-differences model is estimating, but I want to confirm that I am 'translating' it into English/words properly.
Let's say I am running the following event study specification:
$y_{i,t} = \lambda_i +\tau_t + \sum\limits_{k \neq -1}Treat_i *\mathbb{1}\{t=k\}\beta_k + \eta_{i,t}$
where $i$ is the county, $t$ is time, $\lambda_i$ are county fixed effects, $\tau_t$ are time fixed effects, and $Treat_i$ is an indicator = $1$ if a county is in the treatment group, and $\mathbb{1}\{t=k\}$ as an indicator = $1$ if time = $k$. Thus, this is a difference-in-differences, where I replaced a 'post' dummy with a vector of year indicators. I am omitting the interaction $k = -1$, the year prior to the treatment being implemented, as the reference group. $\eta_{i,t}$ is the error term.
For the sake of this example, say $Y$ are mortality rates, deaths per 1000 population.
I want to put the coefficient $\beta_k$ in a sentence. say $k = 9$, and say $\beta_9 = 5$. Is it correct to say:
"Treatment counties experienced 5 more deaths per 1000 than control counties 9 years after treatment relative to the difference in the year before treatment."
Or, can one simply say:
"Treatment counties experienced 5 more deaths per 1000 than control counties 9 years after treatment relative to before."
I am curious if one of those is 'more correct', or are both incorrect?

I am partial to the former interpretation.
Here, $\beta_k$ is interpreted as the effect of treatment for different lengths of exposure to the treatment. As is the convention in most event study frameworks, $\beta_{-1}$ is normalized to be equal to 0. The estimate of $\beta_0$ is the instantaneous treatment effect; it's the average effect in the first year the treatment is implemented. Positive values for $k$ represent treatment lags, which is useful when we want to assess treatment effect dynamics. The estimate of $\beta_9$ is, in effect, the coefficient on the ninth lag. Put simply, it's the average effect of treatment 9 years after the first (immediate) adoption year.
As a general recommendation, don't concern yourself so much with interpreting each lead or lag. Instead, show their evolution over time by plotting the coefficients.
