DiD with time invariant and time variant treatment? I am currently working on a project that seeks to study the impact of the subprime crisis on family firms and non-family firms. In this, I have imagined to use a DiD estimator to account for differences in how family firms, versus non-family firms were affected by the crisis.
However, in this design, both groups receive the treatment, whereas one (family firms) are also subject to a time invariant treatment. Therefore, as I am currently considering it, we essentially have four states:

*

*Pre-crisis, non-family firm (i.e., crisis = 0, family business treatment = 0)

*Pre-crisis, family (i.e., crisis = 0, family business treatment=1)

*Post-crisis, non-family firm (i.e., crisis=1, family business treatment = 0)

*Post-crisis, family firm (i.e., crisis=1, family business treatment = 1)

Where I essentially want to compare the crisis coefficient of 3) and 4).
What I am now asking is: Does it make sense to conduct a DiD in this setting, given that there are essentially two treatments? (Family business, and the subprime crisis). Are there any competing approaches that I am currently overlooking?
I hope that I made myself clear.
Sincerely
Johan Karlsson
 A: 
I essentially want to compare the crisis coefficient of 3) and 4).

You can do this. However, you will only be comparing family firms with non-family firms. Both types of firms ‘turn on’ in the post-period.

Does it make sense to conduct a DiD in this setting, given that there are essentially two treatments?

No, and not because you have more than one treatment. Let me elaborate. The "crisis" (i.e., treatment) is firm-invariant. In other words, you do not have a group of "unexposed" firms. If I understood your post properly, the subprime shock impacts family firms and non-family firms. Note: difference-in-differences requires you to observe at least some unexposed firms over time (i.e., before and after the subprime crisis).
Your second "business" treatment is a time-invariant exposure specific to family firms—only. The problem here is the absence of any pre-treatment observations for family firms. Your family firms are, in essence, always treated. A firm fixed effect would help adjust for this, but then we're back to a mortgage crisis experienced by all firms in your sample. If the subprime crisis impacts a subset of firms at different times, then maybe you could exploit variation in treatment timing. It appears, though, treatment exposure is well-defined, so I don't think you can take this route either. In sum, difference-in-differences does not seem valid in my estimation.
