[Economics] How to justify the treatment and control groups for Difference-In-Difference with staggered implementation of laws?

1. [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.

2. Does it mean that only $()_{}$ of the treatment group after the laws being passed received a value 1? Other than that, the control group and the treatment group before laws being passed receive a value of 0?

Correct.

The binary treatment variable in this more general setting is not the same variable as in the 'classical' difference-in-differences case. Suppose a leniency law is espoused by all firms within treated countries in the year 2000. In this setting you could write this equation more simply as the interaction between a treatment-control dummy and a post-treatment indicator equal to 1 after the law goes into effect in both groups, 0 otherwise. However, once we move away from this setting and the roll out of treatment is staggered or even switching 'on' and 'off' over time, then the "post-treatment" variable is no longer well-defined. To proceed, we must use the 'generalized' difference-in-differences estimator which defines the product term in a different way.

In the equation you reference, the interaction term is implicit in the coding of $LL_{kt}$ (i.e., leniency law). It is equal to 1 in all firms that are headquartered in countries that have passed a leniency law by year $t$, 0 otherwise. Imagine you have a column of zeros. Input a value of 1 once the firms in country $k$ adopt by year $t$, 0 otherwise! Thus, any country never adopting a leniency law is left as 0 for the entire observation period.

3. Imagining two cases: (1) Both Germany and Brazil implemented this law in the year 2000 and (2) Only Germany implemented this law in 2000. So whether the numbers of observations of the control groups for these two cases are identical?

From the perspective of this estimator, the treatment variable $LL_{kt}$ makes no distinction between treatment/control groups. We only have countries $k$ switching on their policies by year $t$. Any country $k$ never adopting a leniency law is equal to 0 in all $t$.

Suppose a panel runs from 1995–2002. In example (A) both Germany and Brazil will 'turn on' (i.e., switch from 0 to 1) in the year 2000 and stay on. Thus, Brazil and Germany have 5 pre- and 3 post-treatment time periods. They have "identical" treatment histories. The countries never espousing the new law would serve as a counterfactual.

In scenario (B) only the within-group observations in Germany would switch on in the year 2000, while the within-group observations in Brazil would all equal 0. The countries never espousing the new law, which now includes Brazil, would serve as a counterfactual.

4. Australia and Belgium passed the law in 2003 and 2004, accordingly. So, firms in Malaysia only exist once in the control sample? I mean, whether one observation can be the control for many treatments in this DiD setting?

Yes.

Suppose Malaysia never adopts a leniency law—ever. Malaysian firms may serve as a counterfactual for Australian and Belgian firms up until the time periods in which they are treated. In other words, the Malaysian sector is a counterfactual for the early- and late-adopters before each of their respective exposure periods.

The data frame below shows how we code $LL_{kt}$ in practice. Note this is a little different since we only see country-year observations. I assume based upon my cursory review of the paper that leniency laws affect all firms within each country. If so, and the authors observe all firms before and after the laws goes into effect, then we can estimate the equation at the firm level or at the country level.

For simplicity, the fictitious data frame below observes three countries from 2000–2006. Germany espouses the law early; $LL_{kt}$ switches from 0 to 1 in 2003 and stays on. Australia adopts late; $LL_{kt}$ switches from 0 to 1 in 2005 and stays on. The Malaysian market never adopts a leniency law. Note how $LL_{kt}$ is 0 for the entire observation period. If we also observed firms $i$ over time within Malaysian industries, then all firms should be coded 0 in every firm-year period.

Again, the variable $LL_{kt}$ is not delineating a specific subset of treated or untreated firms/countries; rather, it simply 'turns on' (i.e., switches from 0 to 1) if a jurisdiction was treated and only during the periods $t$ when the law was actually in effect, 0 otherwise.

The following data frame should help with your intuition:

$$ \begin{array}{ccc} country & year & LL_{kt} \\ \hline \text{Malaysia} & 2000 & 0 \\ \text{Malaysia} & 2001 & 0 \\ \text{Malaysia} & 2002 & 0 \\ \text{Malaysia} & 2003 & 0 \\ \text{Malaysia} & 2004 & 0 \\ \text{Malaysia} & 2005 & 0 \\ \text{Malaysia} & 2006 & 0 \\ \hline \text{Germany} & 2000 & 0 \\ \text{Germany} & 2001 & 0 \\ \text{Germany} & 2002 & 0 \\ \text{Germany} & 2003 & 1 \\ \text{Germany} & 2004 & 1 \\ \text{Germany} & 2005 & 1 \\ \text{Germany} & 2006 & 1 \\ \hline \text{Australia} & 2000 & 0 \\ \text{Australia} & 2001 & 0 \\ \text{Australia} & 2002 & 0 \\ \text{Australia} & 2003 & 0 \\ \text{Australia} & 2004 & 0 \\ \text{Australia} & 2005 & 1 \\ \text{Australia} & 2006 & 1 \\ \end{array} $$

Note, before Germany adopts a leniency statute in 2003, their counterfactual history is the Malaysian and Australian sectors. It is also worth highlighting that previously treated countries may also serve as counterfactuals for the late-adopter countries. In other words, when Australia adopts a leniency law later in 2005, the Malaysian firms (i.e., non-adopters) and the German firms (i.e., early-adopters) may approximate their counterfactual history. This estimator is actually averaging all possible 2x2 difference-in-differences estimates.

The downsides of this estimator may serve as a whole separate discussion. But let's address one concern briefly. For instance, the model assumes the absence of time-varying effects. For example, the effect of a leniency law on Germany's outcome trajectory is assumed to be instantaneous and constant. This assumption is tenuous and difficult to defend statistically. Note how we still assume a "common trend" across all groups before their exposure periods. Imagine a scenario where Germany's outcome trend was changing over time after 2003. Once Australia moves into the treatment condition in 2005, then Germany's outcome trajectory may be offset by their earlier exposure. When 'already-treated' countries are allowed to act as controls for 'soon-to-be-treated' countries, then changes in their treatment effects over time get subtracted from the difference-in-differences estimate. So while the two-way fixed effects estimator is yielding a weighted average of treatment effects across all groups and times—some of the weights may be negative. The negative weighting only arises in the presence of heterogeneous treatment effects. Peruse this working paper for more on this topic.

5. Above is how he grouped the control and treatment based on his identification, but on page 2600, Dasgupta, 2019, section 3.1 said that "control firms are all firms in the same industry in countries that had not passed a leniency law in the 7 years surrounding the event date". This identification confused me because now we have one more dimension about the industry. I am wondering why they did not put it right in the baseline identification but instead in results section? What is the reason behind that?

It is my understanding that the timing of leniency laws did not vary across industries, but I can't be sure until I give the paper a thorough read.

Usually authors may specify a base model and then expand upon it in their results section. For example, they may report lead/lag coefficients in tabular form but do not explicitly show their equation. The authors may also try a series of alternative specifications such as including industry-by-year effects even though this term was excluded from their main empirical equation.