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  • Difference in difference with different start of treatment

    I am running a DiD model for 20 companies in the treatment and 35 in the control. The time goes from 1960 to 2010 and the treatment started for most companies in 1970, but for one in 1972 and for another in 1973....so, the question is how do I specify these in the equation? Thankyou

  • #2
    Something happened in 1970 that led a bunch of companies to undergo the "treatment." For some reason two companies that did not do that, did so a few years later. The key question is, what is it about these two companies that led them to "resist" the 1970 trend and then "succumb" in 1972 and 1973. And then, can you identify something similar in some of the controls?

    To salvage something approximating a DID design you need to identify a control for each of those two firms. In the simple DID design, where all the treatment firms begin the treatment simultaneously, the logic is that the control firms "would have" begun treatment at that same time had they begun treatment at all. In your situation, this logic fails. But perhaps based on the facts of this particular intervention, there is a rational basis for saying that some of the controls, had they adopted the treatment, would have done so in 1972 or 1973. If you can do this, then the technical aspects are fairly simple. To do a DID analysis you need two variables: one is an indicator ("dummy") for being in the treatment or control group. The other is a pre-post indicator. In the classic DID analysis the pre-post indicator would simply distinguish before 1970 from 1970 and beyhond. In this analysis, the prepost variable would, for the intervention group, distinguish before the actual starting date for that firm from after, and for the controls it would distinguish the period before their "would have started" date from after.

    Now, it may be that you cannot identify a "would have started date" for the controls (or that such a would have started date doesn't really correspond to 1970, 1972 or 1973. In that case, you could try to create matched pairs. Pick controls that are more similar in relevant ways to the two oddball firms, and treat them as "would have started" in 1972 and 1973 and set their pre-post indicator variable accordingly.

    If there is no rational basis for creating matched pairs, then you are reduced to the weakest approach of picking two controls at random and imputing to them a 1972 or 1973 "would have started date."

    Finally, since there are only two strange firms, there is another option to consider here. If you had a larger number of firms, the best solution, I think, would be to simply omit those two oddball companies from the study altogether. They are neither treatment nor control companies; they are "not in universe." With only 20 intervention companies altogether, though, you might not be able to afford to discard that data. If the treatment effect you are estimating is a strong one, even with this sample size you might be able to drop these two firms without obliterating it. I will say that even if you end up using one of the approaches mentioned earlier in this response, I would also run an analysis omitting these two firms as a sensitivity analysis.


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