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Experts,

I have some trouble concerning my regression model for a panel data analysis. The dataset includes observations of 200 firms over a period of 6 years (2000 - 2005) regarding merger activities and profit. The aim is to model the impact a merger has on firms' profit. For that reason, a regression with fixed effects should be done. At the moment it looks like that:

Code:

xtreg depvar indepvar i.postmerge##i.treated i.year, fe

With postmerge (1 for years after mergers) and treated (1 for firms involved in a merger (constant over time) as a binary dummy variable. The challenge is, that the year, in which a merger takes place, differs from firm to firm. Therefore, the variable postmerge is 0 for all the firms of the control group over all the time periods.

The problem is, that if I run this regression in Stata, i.postmerge*i.treat and i.treat gets dropped because of collinearity. I suppose that the drop of i.treat is due to the fixed effects (variable treat does not change over time). Now, I have several questions:

  1. If I replace the interaction of postmerge and treated 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)?

  2. Which combination measures the treatment effect the best way while keeping fixed effects in mind?

  3. Most important: What would an appropriate -xtreg- model look like for this estimation?

I appreciate any kind of help.

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  • $\begingroup$ As far as your first question is concerned, you state that the time-constant treatment variable was kept in combination with the year. Am I correct to say ‘treat’ by itself was still dropped? $\endgroup$
    – Thomas Bilach
    Sep 16, 2022 at 14:38
  • $\begingroup$ Exactly! Treat standing alone gets dropped, no matter if the year dummy is included or not $\endgroup$
    – Sam
    Sep 16, 2022 at 14:44

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The problem is, that if I run this regression in Stata, i.postmerge*i.treat and i.treat gets dropped because of collinearity.

This is not a problem.

The variable 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 postmerge and 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.

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  • $\begingroup$ Dear Thomas, I am blessed you take that much time helping me out! Thank you so much! The explanation sounds logical to me, but there are some questions left regarding the 'time_since_merger'-variable. In order to compute it, I am wondering which value the 'time-since-merger'variable should take for firms of the control group? So firms, that never undergone a merger. And what do I have to do to get Stata to include it in regression? I got the error message 'time_since_merger: factor variables may not contain negative values' $\endgroup$
    – Sam
    Sep 17, 2022 at 6:00
  • $\begingroup$ Technically, they’re the “always 0” group because we don’t have a natural year where mergers happen for all firms. In your setting, I see now that you’d end up with negative values for the pre-merger periods. The control group is likely filled with missing values, or a constant negative value (e.g., -5) for the controls. As a clever hack, try adding a positive integer to years_since_merger, then use the label define command to create a map between the numeric values and the words/symbols used to describe the relative years “approaching” a merger (i.e., the “leads”). Does this make sense? $\endgroup$
    – Thomas Bilach
    Sep 17, 2022 at 6:26
  • $\begingroup$ Thank you again! It does make sense and is a good way to deal with it. I replaced 'yeas_since_merger' with 'year_since_merger'+10 and defined labels according to its actual value. Now I got regression results. Unfortunately '1.treat' and 'treat*1.years_since_merger' gets still dropped out of the model. $\endgroup$
    – Sam
    Sep 17, 2022 at 6:59
  • $\begingroup$ You could redefine the sequence where the addition of the constant forces the controls to become 0. As for the redundancies, do you get output for all the relative periods? Are they all omitted? $\endgroup$
    – Thomas Bilach
    Sep 17, 2022 at 7:03
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    $\begingroup$ With i.treat#i.years_since_merger only 2 out of 9 periods get omitted. Actually, I don't unterstand the difference, because in my thinking, the only difference between # and ## is, that both indepvars gets additionally displayed standing alone. However, I am happy it works out now. Thanks again! The last question refers to the kind of interpretation. $\endgroup$
    – Sam
    Sep 17, 2022 at 7:32

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