Why the standard three-way interaction term isn't going to work in staggered DiD? I did not have enough reputation to comment so I am asking a thing I am confused about here. I am learning triple diff (DDD) from this topic from Thomas
In this topic, he quoted:

It's important to be aware that because the laws are introduced at different times, a standard three-way interaction term isn't going to work.

I do not understand why the standard three-way interaction term cannot work in this case. And does it mean that in a setting where the event date is invariant to the group that you are in, then we can consider using the standard three-way interaction term?

To be clear, when I say a "standard three-way interaction term" I am saying that the product of three terms isn't going to create the policy (i.e., treatment) variable for you.
In the 'classical' case, we typically work with three variables of interest: a dummy for group status (i.e., treatment/control dummy), a dummy indicating pre- versus post-treatment (i.e., pre/post indicator), and a dummy for the more sensitive group nested within units (e.g., young/old). Assuming all units adopt the new law at the same time, then a product term would instantiate the treatment variable appropriately (e.g., $Treatment_i \times Post_t \times Age_a$). In software, for example, all of the lower order interaction terms would be estimated for you—for free.
In R, for example, a "standard" triple interaction term looks like this:
lm(Y ~ T*P*A, data = ...)

which is actually the same as estimating:
lm(Y ~ T + P + A + T*P + T*A + P*A + T*P*A, data = ...)

In your setting, however, the term P (i.e., post-treatment) has no well-defined meaning. Some units enact new legislation much earlier than others, and some don't consider adopting at all. We cannot assume the untreated would have passed the new law at any one particular time period, because the passage of the new law is starting at different times for different units. As a consequence of the staggered nature of treatment, the individuals/firms/industries nested within your larger aggregate units will also be affected by the law at different times. This doesn't mean you cannot estimate a difference-in-difference-in-differences equation, it just can't be achieved by simply interacting a few variables and expecting software to return your treatment effect. Instead, you must instantiate the law dummy manually to account for the different post-periods. If executed properly, the three-way interaction term is implicit in the coding of the law dummy.
