What are the dynamic and static event-study?

Question

When reading about the Difference-in-Differences for multiple groups and time periods, I see a term called "static or dynamic/event-study". Can you help me to distinguish these two settings intuitively? I tried to google but I cannot find any intuitive answer.

Static event study

$\begin{equation} y_{it} = \mu_i + \mu_t + \tau D_{it} + \varepsilon_{it}, \end{equation}$ (1)

where $\mu_i$ are unit fixed effects, $\mu_t$ are time fixed effects, and $D_{it}$ is an indicator for receiving treatment.

Dynamic event study

$\begin{equation} y_{it} = \mu_i + \mu_t + \sum_{k = -L}^{-2} \tau^k D_{it}^k + \sum_{k = 1}^{K} \tau^k D_{it}^k + \varepsilon_{it}, \end{equation}$ (2)

where $D_{it}^k$ are lag/leads of treatment (k periods from initial treatment date). Side question: what does L mean in the second equation?

Reference updated

Answer 1

The first model you presented is for a case where the treatment is applied, the effect occurs, and then no further effects from the treatment are observed.

The second model imagines that the treatment could be preceded by some sort of anticipation effect (such as investors anticipating that the Fed will change interest rates). The maximum number of periods this anticipation could take before the treatment occurs is L (leads). Similarly, some effects could be delayed (some investors didn't check the newspaper until the following week). These delayed effects can occur up until K periods later (lags).

Empirical work in my area tends to recognize that lags and leads are plausibly relevant, and in the case that the first model is true and there are no confounding dynamics, all the lags and leads should be zero. However, in practice there seems to often be some significance to these lags and leads even if the circumstances seem to suggest an event best described by the first model.