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In a topic asked, the expert mentioned the word "counterfactual"

[W]hy do they need to write down "adopted a leniency law at some later point of time"? Because in Korea case, the word "our sample period" means "1995-2002" already.

=> Assuming Korea is the early-adopter country, then all countries theretofore untreated before 1997 may serve as a counterfactual. This includes the countries never adopting a leniency law and those with impending treatment adoption periods.

I did some searches around but I still cannot clarify this word, I am wondering if it mean "control group" in the Difference-in-Difference setting.

And whether he means "all countries passed the laws before 1995, or after 2002, or never adopt the laws all belong to the control group?

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    $\begingroup$ If you had complete control of the space-time continuum, you could assess the effect of your policy on outcomes in Korea and then go back in time and reassess outcomes in Korea in the absence of the policy. Unfortunately, we can't go back in time and observe both conditions. In this universe, we only know one possible treatment history for Korea. The factual state of affairs is that Korea was exposed to a treatment. The other untreated countries may represent the "counterfactual" state of affairs, which we use to approximate what would have happened had Korea not been exposed to treatment. $\endgroup$
    – Thomas Bilach
    May 22, 2021 at 18:42
  • $\begingroup$ @ThomasBilach thank you very much, the more I read your comment, the more I understand the empirical setting. I am reading the working paper given by you and read your comment more carefully and will reply or start a chat with you soon in another bountied topic. Thank you so much for your help so far. $\endgroup$
    – Phil Nguyen
    May 22, 2021 at 22:02

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In order to do causal inference, we need to have a counterfactual. In the case above it would be the outcome variable in Korea had the leniency law not passed. However, this is fundamentally impossible to learn. So instead we try to find a suitable control group that could "simulate" what that counterfactual might look like. This simulation is part of why we need to justify the parallel trend assumption, we essentially want to make sure that the outcome variable in our control group will mirror the outcome variable in the treatment group had the treatment never occurred.

Formally we can use potential outcomes notation to describe this,

$$Y_{i,D=0}$$ vs. $$Y_{i,D=1}$$

where we would like to know the outcome $Y$ for individual $i$ in the case of treatment $D=1$ and control $D=0$.

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