# 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?

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.
• Thank you for this intuitive answer. I am taking your recommendation into account, but just for my own intuition to clarify something, does that mean the effect for $\beta_9$ is the \textit{cumulative} effect 9 years afterwards? in other words, if we assume there is an underlying independent effect for each year, it would yield the sum of each of these effects up to 9 years? Commented Sep 15, 2021 at 1:51