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In my difference in differences model firms $> x$ belong to the treatment group whereas firms$< x$ act as control.

I have a two period model:

  • In $t_1$ firm $i$ is $> x$ and thus belongs to the treatment group
  • In $t_2$ the same firm $i$ is $< x$ and belongs to the control group

Do you have any recommendations how to deal with this problem?

Additional information:

The treatment is a change in a law that only affects firms with more than $x$ employees. Firms smaller than $x$ are not affected by this law. However, a firm that is $<x$ in $t_1$ can certainly exceed the threshold in $t_2$. In this case it will be affected by the law change in $t_2$

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Sometime this is referred to as treatment cross-over (unintentional in this design, although it can be intentional in some randomized studies). One approach is to utilize instrumental variables and estimate the ITT (Intent to Treat) effect as opposed to the effect of the actual treatment assigned. I suspect given your panel design though there is a more direct way to model the treatment. What is the treatment exactly? –  Andy W Sep 10 '12 at 15:40
    
What is the treatment exactly? A change in law ( I have added this information to the question now) –  Julian Sep 10 '12 at 16:39
    
At first blush, this sounds more like two regression discontinuity designs to me, one for each period. It would be nice to get the same answer in both periods. On the other hand, since $x$ is chosen by the firms, RD would not be estimating the effect of the law if firms lay off workers to avoid compliance. –  Dimitriy V. Masterov Sep 10 '12 at 17:11
    
@DimitriyV.Masterov: yes, with two periods RD might be an alternative. However, If I had multiple periods I would assume a common trend and use DD instead. Concerning x, that x is chosen by the firms is also a problem in a DD framework –  Julian Sep 10 '12 at 18:15
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