What shoud we do when the expected treatment overlaps control sample in DID?

Question

In a generalized Difference-in-Differences in Dasgupta,2019 paper, he documented that

control countries did not have a leniency law introduced over 2 years before to 5 years after the introduction of the leniency law for the treated countries.

And it is how he describe sample selection

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

There is an example, Brazil implemented the law in 2000 and United States implemented the law in 1993. U.S implement the laws in 1993 means that Brazil in 1998 is control for US (instead of using firms in Brazil, I used Brazil here for shortening purposes). However, Brazil also passed the law in 2000, meaning that 1998 also belong to treatment sample.

So, what should we do in this case? In particular, which sample (control or treatment) that Brazil in 1998 belongs to?

Another example, Lithuania passed the law in 2008, Ukraine passed the law in 2017,so whether the firms in Lithuania from 2014 to 2017 is the control of Ukraine (because Lithuania from 2006->2013 are treatments). It makes sense if based on the switch on and off of Thomas from this discussion. However, on the other hand, it makes no sense, because the control for Ukraine cannot compared to the change of Lithuania before and after 2017. More detail about this example can be seen from the explanation here.

Answer 1

Your first quote seems to be given as an explanation of how they constructed Figure 1 (however I cannot be sure since you did not state pagenumber for the quote). Anyway, Figure 1 compares a treatment group with a control group on outcome variable. The descriptive content of these figures are not rigorously related to the main estimation equation

$$(1) \ \ Y_{it} = \alpha + \beta (Leniency Law)_{k(i),t} + \delta X_{it} + \theta_t + \gamma_i +\epsilon_{it},$$

where $k$ is country, $i$ is firm and $t$ time.

The estimator $\hat \beta_{OLS}$ do not presuppose existence of control and treatment group there is no group indicator. All that is presupposes is $(Leniency Law)_{k(i),t}$ implying that for any time $t$ you know which country $k(i)$ firm $i$ is in and whether leniency laws were adopted.

When econometricians still talk about treatment and control groups when they apply the estimator $\hat \beta_{OLS}$ it is because the believe they can interpret $\hat \beta_{OLS}$ somewhat like it is possible to interpret $\hat \beta_{OLS}^{2\times 2}$ where $\hat \beta_{OLS}^{2\times 2}$ is defined by estimation eqaution (1) in the case of 2 time periods and 2 countries - one adopting leniency laws one not.

This is all summarized clearly in Goodman and Bacon (2018) and in this blog post.

Obviously, econometricians are interested in causal estimates. One main tool for theorizing about the causal estimates are the Rubin Causal Model. There exists a literature trying to apply this framework to the interpretation of $\hat \beta_{OLS}$ in setting going beyond the $2 \times 2$ setting.

The main insight of this literature is that $\hat \beta_{OLS}$ can be seen as a weighted average of treatment effects Chaisemartin & D'Haultfæuille (2020). And since these particular treatment effects can be estimated by a proper choice of 2 x 2 design this can be seen as a step in direction of interpreting $\hat \beta_{OLS}$ in these terms. However, these interpretations are not performed trivially and are always based on

(1) Specifics of the datastructure (such as staggered design for Goodman and Bacon)

(2) Assumptions imposed on the used RCM model

and to my understanding they do not in general imply that for example any single country can be placed in treatment or control group as you seem to be trying to do.

The simplest of problems is how to expands to a framework with multiple timeperiods but where the two observed groups can still be split up into treated and non-treated in a simple fashion because all treated start being treated at the same time see for example here for a good intro to treatment effects with multiple periods.

So to put it in short: So, what should we do in this case? In particular, which sample (control or treatment) that Brazil in 1998 belongs to? If you wanna estimate equation (1) you should do nothing, this is not a problem for that estimator. It may be a problem for a particular application of an RCM model but that remains to be shown.