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I follow Dasgupta, 2019 to set up the control and treatment variables for staggered implementation of laws among countries as events. The treatment and control groups are explained in one topic.

I am wondering if there is any reference paper or any explanation guiding how to perform the pre-trend analysis (equation) for this case. I did some searches and saw this topic. However, the answer from the expert not mention the econometrics way or the equation to test pre-trend analysis.

In Dasgupta,2019, last paragraph, section 4.2, I saw a paragraph denoting how he conduct the pre-trend test

To explore the dynamics of the issuance activities and leverage change, we create dummy variables corresponding to the following windows around the treatment year: from 1 to 4 years before the treatment; the treatment year and the 2 years after treatment; the next 3 years; and the years beyond. We find that firms first start growing and issuing equity over the first two sub-periods after the discussion about the leniency law passage started and there is no pre-trend once anticipation effect is taken into account.

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So you typically need at least two pre-periods for both treatment and control in order to consider pre-treatment trends. Based on my review of the paper it seems that they do indeed have two (please correct me if that is incorrect!).

The intuition of testing pre-trends is simple. Let us use potential outcomes notation where $Y_0$ refers to the non-treated potential outcome and $Y_1$ refers to the treated potential outcome. The intuition is that during the pre-treatment periods any observed change is a change in $Y_0$.

Often people will just eyeball this by plotting the outcome over periods prior to treatment and periods post-treatment. Figure 1 seems to do this in the paper.

A more formal test could also be concocted. What you could do here is to drop all periods post-treatment and regress the outcome, $Y$, on group dummies, time dummies, and group-time dummies. If the coefficient on group-time dummies is not significant this can serve as a decent test of pre-trends. The idea here is almost identical to the DID in that we are controlling for within-group and within-time variation and so what we get is variation within both group and time before treatment. So if there already was some strong pre-treatment trend we should pick it up.

Though to be frank, I am not sure people bother to do a formal test like this in practice. Every DID paper I have seen usually just relies on the visual test. The reason for this is simply that in almost any practical case a significant pre-treatment trend should appear in a visual evaluation. I think if the visual inspection is concerning then using the formal examination may prove useful. Garthwaite et al. 2014 (Section IV.D) is a good example of using the visual test to check for pre-trends.

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  • $\begingroup$ Thank you for your dedicated answer. I am wondering one part "What you could do here is to drop all periods post-treatment and regress the outcome, Y, on group dummies, time dummies, and group-time dummies". What is the group dummies., time dummies, and group-time dummies here then? Are they the whole sample of control and treatment concatenating together during the pre-treatment periods? $\endgroup$
    – Louise
    May 31 '21 at 5:06
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    $\begingroup$ These dummies are just indicators of control and treatment (groups), time periods prior to treatment, and their interaction. Yes you would concatenate the pre-treatment periods for this exercise. $\endgroup$
    – Ariel
    May 31 '21 at 12:38
  • $\begingroup$ excuse me, I am surprising about "their interaction" here, what does this mean during the pre-treatment period then? Thank you. $\endgroup$
    – Louise
    May 31 '21 at 22:55
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    $\begingroup$ Sorry if this is unclear, I mean you should use two-way fixed effects and you should be checking for significance for the two-way dummies. $\endgroup$
    – Ariel
    May 31 '21 at 23:00

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