The problem is, that if I run this regression in Stata,
i.treat gets dropped because of collinearity.
This is not a problem.
treat, as you have defined it, is a time-constant variable indicating a firm's membership to the treatment group. Since it doesn't vary over time, it is collinear with the firm fixed effects. Software will exclude it.
If I replace the interaction of
treat with an interaction like
i.year#i.treat, Stata keeps the variable treat in combination with the year. Does this makes sense and how can I interpret the results in terms of an overall result (Results are one coefficient per year)?
It makes sense that software is keeping the product terms, though
treat will be dropped, for reasons already mentioned. However, the interpretation is going to be a bit murky, since mergers are happening at different times. To make sense of this, we need to know the relative time to a firm's first exposure. So while you can interact your treatment dummy with a variable denoting calendar year, it still doesn't tell us how treatment varies with time since exposure.
Which combination measures the treatment effect the best way while keeping fixed effects in mind?
The best way is to create a combination of treatment with relative time. For example, say firm $A$ merges in 2003. The immediate effect is a dummy equal to 1 for firm $A$ in 2003, 0 otherwise. The first lag just repeats this process, where now firm $A$ equals 1 in 2004, 0 otherwise. Note how this is the interaction terms you're referring to, just defined in a different way. It's a combination of treatment with relative time.
Most important: What would an appropriate
-xtreg- model look like for this estimation?
There are quite a few computationally efficient ways of generating time-varying treatment effects without manually creating all the leads and lags of the merger variable. I don't have any of your data to work with, so I can only guide you in the right direction. Assuming mergers never happen for a subset of firms, then all you need is some way of delineating the relative periods for those firms actually experiencing a merger. You already have a variable which distinguishes the treatment group from the control group. As a next step, generate an "event time" variable, call it
years_since_merger, which differences a firm's first merger year from calendar year.
firm year first_merge years_since_merger
A 2000 2003 -3
A 2001 2003 -2
A 2002 2003 -1
A 2003 2003 0
A 2004 2003 1
A 2005 2003 2
You can think of this computation as centering all firms around that first merger year.
xtset firm year
xtreg profit i.treat#i.years_since_merger other_covariates i.year, fe cluster(firm)
Now the interactions are directly interpretable. Before, you were conflating calendar time with event time. In my fake example, the immediate merger year for firm $A$ is 2003. Note how in 2001 firm $A$ is two years before the event. But for another firm, say firm $B$, that could be its actual merger year.
In short, by creating a variable which denotes event time, we more precisely exploit the timing impact of firm mergers.