Recently, I read two papers by Dasgupta,2019 and Dong,2019 examining the impact of staggered laws in different dependent variables. When having a really long discussion with people in the comment section here, I accidentally recognized that these two papers use different treatment/control samples, which seems to against the well-explained idea about the switching "on" and "off" of treatment-or-control status based on treatment year shown here (this is where the baseline regression being explained well) and here.
In short, @ThomasBilach's idea is:
Once the exposure period isn't well-defined we must regress the outcome on unit fixed effects, time fixed effects, and a treatment dummy which 'turns on' in any unit-time combination where the policy is in effect, 0 otherwise.
This is how Dasgupta, 2019 explains the sample selection in his paper in the following way:
the treated group comprises all firms that are headquartered in countries that have passed a leniency law by year t. The control group comprises firms in countries that never adopted a leniency law in our sample period and firms headquartered in countries that adopted a leniency law at some later point of time.
Although I am doubting whether we can plot a pre-trend analysis based on this description, however this explanation seems to align with Thomas Bilach's idea as above.
However, Dong, 2019 documented that
In our estimation, we rely on the staggered nature of the passage of leniency programs to identify their causal effect on ﬁrm margins. We follow the standard approach used in the literature, which relies on the staggered passage of laws in different geographic regions in US. This allows us to compare the change in the margins of ﬁrms that were affected by the law to the contemporaneous change in the margins of the control ﬁrms that were headquartered in countries that had not yet passed such a law.
This description is different the description by Dasgupta, 2019, and as a result, does not align with Thomas Bilach's description.
To be simplified: from Thomas Bilach's idea and Dasgupta, 2019, if the U.S. passes a laws in 1993 and Australia passes a law in 2004, it means that the U.S. can be a control for Australia, but this is not the case based on the description of provided by Dong, 2019.
So the main question is, it seems that Dong, 2019 is using the staggered DID or generalized DID but they seem not to follow this method. I am doubting if I am misinterpreting them because the two papers Dasgupta, 2019 and Dong, 2019 have the same author (Alminas).