regression difference in differences with multiple treatment period and Granger test

Question

I'm currently working on an unbalanced panel data which contains 10000 + individuals and at most (at least) ten (five) years observations. A type of treatment happens to some of the individuals but at different time (multiple treatment periods), except the treatment effect, I also want to test if consequence happen before treatment (Granger test, Granger 1969). There is a reference in Mostly Harmless about this problem, also in this post with equation like this : $$ \text{Y}_{ist}=\gamma_s+\lambda_t+\sum^m_{\tau=0}\delta_{-\tau}\text{D}_{s,t-\tau}+\sum^q_{\tau=1}\delta_{+\tau}\text{D}_{s,t+\tau}+\text{X}'_{ist}\beta+\varepsilon_{ist} \tag1$$

However, it's unclear neither in the Mostly Harmless nor the reference that how to specify the second term in the equation, cause in the standard DID, we have:

$$y_{it}=\alpha_0+\alpha_1\text{Treat}_i+\alpha_2\text{After}_t+\delta(\text{Treat*After})_{it}+x'_{it}\beta+FFE+TFE+\varepsilon_{it} \tag2$$

Where Treat * After equal "D" in the first equation.

So should we include an "After" for each of the 'D' in equation (1), or something else? Or, how to construct the second term in equation (1) anyway?

Answer 1

This is a very old question, but I might as well answer it.

The leads and lags in the dynamic DiD TWFE shown in Mostly Harmless Econometrics should not be understood like a lagged variable that you now from time series regression. That is, for a given observation we don't use the value of the treatment indicator 5/4/3/... periods ago.

Instead, what we do is that we interact the treatment indicator with a dummy that equals 1 if the observation is 5/4/3/... periods before(after) treatment for the lead(lag).

All pre-treatment indicators should then be zero if parallel trends hold and we have no anticipation effects.