<p>I think the terms &quot;treatment group&quot; and &quot;control group&quot; are at best a loose analogy in an econometric model with two way fixed effects and staggered adoption of the treatment group. In short, I think you are right to be skeptical of the use of the terms &quot;treatment group&quot; and &quot;control group&quot; here. I think the authors are using them in a less way that is less precise or rigorous way to give a general intuition for the identification strategy by way of an analogy to a randomized trial or a non-staggered difference-in-difference.</p>
<p>There is actually a pretty big recent literature about econometric issues that are somewhat related to this, especially in cases where the treatment effect is not uniform. If you search for &quot;difference in differences&quot; on Twitter you can learn a lot about it.</p>

CC BY-SA 4.0
2021-05-12T09:43:54.707Z
Thank you @Jonathan Borowsky, I thought you went through this paper a little bit already, I much appreciate it if you can give your own idea about my 5 curiosities above. I did search in Twitter but the knowledge seems not to be concentrated and there is no place to ask. Warm regards and thanks
Phil Nguyen
CC BY-SA 4.0
2021-05-12T01:30:47.937Z
2021-05-12T01:30:47.937Z
Jonathan
<blockquote>
<ol>
<li>[W]hy do they need to write down &quot;adopted a leniency law at some later point of time&quot;? Because in Korea case, the word &quot;our sample period&quot; means &quot;1995-2002&quot; already.</li>
</ol>
</blockquote>
<p>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 <em>and</em> those with impending treatment adoption periods.</p>
<blockquote>
<ol start="2">
<li>Does it mean that only <span class="math-container">$()_{}$</span> 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?</li>
</ol>
</blockquote>
<p>Correct.</p>
<p>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 <em>and</em> a post-treatment indicator equal to 1 after the law goes into effect in <strong>both groups</strong>, 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 &quot;post-treatment&quot; 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.</p>
<p>In the equation you reference, the interaction term is implicit in the coding of <span class="math-container">$LL_{kt}$</span> (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 <span class="math-container">$t$</span>, 0 otherwise. Imagine you have a column of zeros. Input a value of 1 once the firms in country <span class="math-container">$k$</span> adopt by year <span class="math-container">$t$</span>, 0 otherwise! Thus, any country never adopting a leniency law is left as 0 for the entire observation period.</p>
<blockquote>
<ol start="3">
<li>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?</li>
</ol>
</blockquote>
<p>From the perspective of this estimator, the treatment variable <span class="math-container">$LL_{kt}$</span> makes no distinction between treatment/control groups. We only have countries <span class="math-container">$k$</span> switching on their policies by year <span class="math-container">$t$</span>. Any country <span class="math-container">$k$</span> never adopting a leniency law is equal to 0 in all <span class="math-container">$t$</span>.</p>
<p>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 &quot;identical&quot; treatment histories. The countries never espousing the new law would serve as a counterfactual.</p>
<p>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.</p>
<blockquote>
<ol start="4">
<li>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?</li>
</ol>
</blockquote>
<p>Yes.</p>
<p>Suppose Malaysia never adopts a leniency law—ever. Malaysian firms may serve as a counterfactual for Australian <em>and</em> Belgian firms up until the time periods in which they are treated. In other words, the Malaysian sector is a counterfactual for the early- <em>and</em> late-adopters before each of their respective exposure periods.</p>
<p>The data frame below shows how we code <span class="math-container">$LL_{kt}$</span> 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 <em>and</em> after the laws goes into effect, then we can estimate the equation at the firm level <em>or</em> at the country level.</p>
<p>For simplicity, the fictitious data frame below observes three countries from 2000–2006. Germany espouses the law early; <span class="math-container">$LL_{kt}$</span> switches from 0 to 1 in 2003 and stays on. Australia adopts late; <span class="math-container">$LL_{kt}$</span> switches from 0 to 1 in 2005 and stays on. The Malaysian market <strong>never adopts a leniency law</strong>. Note how <span class="math-container">$LL_{kt}$</span> is 0 for the entire observation period. If we also observed firms <span class="math-container">$i$</span> over time within Malaysian industries, then all firms should be coded 0 in every firm-year period.</p>
<p>Again, the variable <span class="math-container">$LL_{kt}$</span> 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 <em>and</em> only during the periods <span class="math-container">$t$</span> when the law was actually in effect, 0 otherwise.</p>
<p>The following data frame should help with your intuition:</p>
<p><span class="math-container">$$
\begin{array}{ccc}
country &amp; year &amp; LL_{kt} \\
\hline
 \text{Malaysia} &amp; 2000 &amp; 0 \\
 \text{Malaysia} &amp; 2001 &amp; 0 \\
 \text{Malaysia} &amp; 2002 &amp; 0 \\
 \text{Malaysia} &amp; 2003 &amp; 0 \\
 \text{Malaysia} &amp; 2004 &amp; 0 \\
 \text{Malaysia} &amp; 2005 &amp; 0 \\
 \text{Malaysia} &amp; 2006 &amp; 0 \\
\hline
 \text{Germany} &amp; 2000 &amp; 0 \\
 \text{Germany} &amp; 2001 &amp; 0 \\
 \text{Germany} &amp; 2002 &amp; 0 \\
 \text{Germany} &amp; 2003 &amp; 1 \\
 \text{Germany} &amp; 2004 &amp; 1 \\
 \text{Germany} &amp; 2005 &amp; 1 \\
 \text{Germany} &amp; 2006 &amp; 1 \\
\hline
 \text{Australia} &amp; 2000 &amp; 0 \\
 \text{Australia} &amp; 2001 &amp; 0 \\
 \text{Australia} &amp; 2002 &amp; 0 \\
 \text{Australia} &amp; 2003 &amp; 0 \\
 \text{Australia} &amp; 2004 &amp; 0 \\
 \text{Australia} &amp; 2005 &amp; 1 \\
 \text{Australia} &amp; 2006 &amp; 1 \\
\end{array}
$$</span></p>
<p>Note, before Germany adopts a leniency statute in 2003, their counterfactual history is the Malaysian <em>and</em> Australian sectors. It is also worth highlighting that <strong>previously treated countries may also serve as counterfactuals for the late-adopter countries</strong>. In other words, when Australia adopts a leniency law later in 2005, the Malaysian firms (i.e., non-adopters) <em>and</em> the German firms (i.e., early-adopters) may approximate their counterfactual history. This estimator is actually averaging all possible 2x2 difference-in-differences estimates.</p>
<p>The downsides of this estimator may serve as a whole separate discussion. But let's address one concern briefly. For instance, the model <em>assumes</em> 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 &quot;common trend&quot; 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 <em>weighted average</em> 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 <a href="https://www.nber.org/system/files/working_papers/w25018/w25018.pdf" rel="nofollow noreferrer">working paper</a> for more on this topic.</p>
<blockquote>
<ol start="5">
<li>Above is how he grouped the control and treatment based on his identification, but on page 2600, Dasgupta, 2019, section 3.1 said that &quot;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&quot;. 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?</li>
</ol>
</blockquote>
<p>It is my understanding that the <em>timing</em> of leniency laws did not vary across industries, but I can't be sure until I give the paper a thorough read.</p>
<p>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.</p>

CC BY-SA 4.0
2021-05-20T23:41:58.873Z
thank you for your comprehensive answer, I have a couple of questions as below, much appreciate if you could have a look on that. Regarding the first answer, so, if US enforced the law in 1993, whether US firms from 1995 to2002 is included in the control sample (or counterfactual sample) for Korean ? Even you bold the sentence is the answer 4 but I want to cross check one more time
Phil Nguyen
CC BY-SA 4.0
2021-05-20T23:44:46.967Z
Secondly, I did not get the idea of this sentence, "If we also observed firms i over time within Malaysian industries, then all firms should be coded 0 in every firm-year period.", could you help you to clarify it please? I am wondering why you bring the "industry" up here.
Phil Nguyen
CC BY-SA 4.0
2021-05-21T01:36:55.193Z
Thirdly, I may misunderstand you in this point:"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 research used the window [-2;+5] for their Difference-in-Difference setting. So if Australia adopts leniency in 2005, Germain only be the counterfactual if it passed the laws before 2003. The sample period in section B in my question is [-2;+5] - two years before and five years after the event day. Please correct me if I fall into a fallacy.
Phil Nguyen
CC BY-SA 4.0
2021-05-21T01:39:24.630Z
Fourthly, whether "industry-by-year effects" is running firm fixed effect along with the interaction of year*industry fixed effect, please?
Phil Nguyen
CC BY-SA 4.0
2021-05-21T04:15:56.640Z
(1) Yes. Previously treated countries may serve as counterfactuals for countries treated later. See the paper I referenced which is a detailed treatise on the two-way fixed effects estimator. (2) The fake data frame I provided did not include any data at the $i$-level. The only point I was making is that if you observe firms *within* Malaysia then the treatment dummy is coded 0 for all firms in every single time period.
Thomas Bilach
CC BY-SA 4.0
2021-05-21T04:22:04.273Z
(3) I am under the impression that the firm level data is between 1990–2012. But note that in 'event study' frameworks, we may only consider a finite number of leads and lags around the event date. Suppose Korea's first adoption year is 2000 and Germany's is 2003. The period before exposure (e.g., $-1$) is the year 1999 in Korea and 2002 in Germany. Remember these are relative period effects around the event. Review my answer [here](https://stats.stackexchange.com/questions/523032/testing-parallel-trends-in-staggered-difference-in-differences/523361#523361) for a thorough discussion of this.
Thomas Bilach
CC BY-SA 4.0
2021-05-21T04:27:40.270Z
(4) Yes. The industry * year fixed effects may be included with the firm fixed effects. The authors estimate, separately, industry * year fixed effects *and* region * year fixed effects, both of which were estimated with the firm fixed effects. I believe this was included in Table 3 of Dasgupta's work.
Thomas Bilach
CC BY-SA 4.0
2021-05-21T04:29:17.053Z
Let's jump into a chat if anything else is unclear!
Thomas Bilach
CC BY-SA 4.0
2021-05-23T04:49:48.903Z
Hi @Thomas Bilach, could you please help me to start a chat room, I do not have enough 100 reputations to initiate one chat room(https://stackoverflow.com/help/privileges/chat-rooms#:~:text=How%20do%20I%20create%20a,bottom%20of%20the%20Rooms%20page.). Thank you.
Phil Nguyen
CC BY-SA 4.0
2021-05-23T05:41:57.863Z
Let's see if this works: Enter [here](https://chat.stackexchange.com/rooms/info/124582/difference-in-differences-methods?tab=access.).
Thomas Bilach
CC BY-SA 4.0
2021-05-18T22:09:19.003Z
2021-05-20T15:19:02.480Z
2021-05-20T15:19:02.480Z
Thomas Bilach
<p><strong>A. Background:</strong></p>
<p><a href="https://www.mendeley.com/catalogue/8889aa48-dbef-303a-8896-2880d0d6376d/?utm_source=desktop&amp;utm_medium=1.19.8&amp;utm_campaign=open_catalog&amp;userDocumentId=%7B1df87455-19cc-4494-af93-e274716a758e%7D" rel="nofollow noreferrer">Dong, 2019</a> and <a href="https://academic.oup.com/rfs/article/32/7/2587/5079300" rel="nofollow noreferrer">Dasgupta, 2019</a> used the same way to generate the treatment and control groups because they all learn about the impact of the same laws on different dependent variables. I retrieved the information on how <a href="https://academic.oup.com/rfs/article/32/7/2587/5079300" rel="nofollow noreferrer">Dasgupta, 2019, p. 2596-2597</a> create their sample as below:</p>
<p>The baseline identification:</p>
<p><span class="math-container">$Y_{it}$</span> = <span class="math-container">$\alpha$</span> + <span class="math-container">$\beta$</span> <span class="math-container">$(Leniency Law)_{kt}$</span> + <span class="math-container">$\delta$$X_{ikt}$</span> + <span class="math-container">$\theta$$_t$</span> + <span class="math-container">$\gamma$$_i$</span> +<span class="math-container">$\epsilon$$_{it}$</span> <em>(1)</em></p>
<p>where <span class="math-container">$i$</span>, <span class="math-container">$k$</span>, and <span class="math-container">$t$</span> index firms, countries, and years respectively. <span class="math-container">$X_{ikt}$</span> is a vector of the different firm, country, and industry control, while <span class="math-container">$\gamma$</span> and <span class="math-container">$\theta$</span> are firm and year fixed effects.</p>
<p>The variable of interest here is <span class="math-container">$(Leniency Law)_{kt}$</span>. <a href="https://academic.oup.com/rfs/article/32/7/2587/5079300" rel="nofollow noreferrer">Dasgupta, 2019, p. 2597</a> documented that this variable equals 0 before the passage of the leniency law in country <span class="math-container">$k$</span>, and 1 afterward. They follow a standard difference-in-difference approach with staggered implementation of laws (<a href="https://www.jstor.org/stable/10.1086/376950?seq=1#metadata_info_tab_contents" rel="nofollow noreferrer">Bertrand, 2003</a>) to construct the treatment and control group.</p>
<p><strong>B. Author's description:</strong></p>
<p><a href="https://academic.oup.com/rfs/article/32/7/2587/5079300" rel="nofollow noreferrer">Dasgupta, 2019, p. 2597</a> said that</p>
<blockquote>
<p>the <em><strong>treated group</strong></em> comprises all firms that are headquartered in
countries that have passed a leniency law by year t. The <em><strong>control
group</strong></em> comprises firms in countries that <strong>never adopted</strong> a
leniency law in our sample period and firms headquartered in countries
that <strong>adopted a leniency law at some later point of time</strong>.</p>
</blockquote>
<p><strong>C. Discussion:</strong></p>
<ol>
<li><p>Whether the control group includes firms in countries that implemented this law at some <strong>earlier point in time</strong>? My understanding is: for example, from <a href="https://academic.oup.com/rfs/article/32/7/2587/5079300" rel="nofollow noreferrer">Table 2, Dasgupta, 2019, p. 2599</a>, Korea passed the laws in 1997, therefore, the control group including all firms in all countries not implementing this law from 1995 to 2002, including the US.
From my point of view, it should be the case because based on this identification, firms in the US will not be affected by the laws passed in Korea based on this identification strategy so these US firms from 1995-1996 and 1998-2002 can be control observations for Korea. But if this is the case, <strong>why do they need to write down &quot;adopted a leniency law at some later point of time&quot;</strong>? Because in Korea case, the word &quot;our sample period&quot; means &quot;1995-2002&quot; already.</p>
</li>
<li><p>Does it mean that only <span class="math-container">$(Leniency Law)_{kt}$</span> of the treatment group after the laws being passed received a value 1? Other than that, the control group and the treatment group before the laws being passed receive a value of 0?</p>
</li>
<li><p>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?</p>
</li>
<li><p>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? (meaning no duplicate control observation).</p>
</li>
<li><p>Above is how he grouped the control and treatment based on his identification, but on page 2600, <a href="https://academic.oup.com/rfs/article/32/7/2587/5079300" rel="nofollow noreferrer">Dasgupta, 2019, section 3.1</a> said that &quot;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&quot;. 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?</p>
</li>
</ol>

<econometrics><international-economics><definition><paneldata><corporate-finance>
How to justify the treatment and control groups for Difference-In-Difference with staggered implementation of laws?
Phil Nguyen
# How to justify the treatment and control groups for Difference-In-Difference with staggered implementation of laws?

(Asked by: username_0 on 2021-05-10)

**A. Background:**

[Dong, 2019](https://www.mendeley.com/catalogue/8889aa48-dbef-303a-8896-2880d0d6376d/?utm_source=desktop&utm_medium=1.19.8&utm_campaign=open_catalog&userDocumentId=%7B1df87455-19cc-4494-af93-e274716a758e%7D) and [Dasgupta, 2019](https://academic.oup.com/rfs/article/32/7/2587/5079300) used the same way to generate the treatment and control groups because they all learn about the impact of the same laws on different dependent variables. I retrieved the information on how [Dasgupta, 2019, p. 2596-2597](https://academic.oup.com/rfs/article/32/7/2587/5079300) create their sample as below:

The baseline identification:

$Y\_{it}$ = $\alpha$ + $\beta$ $(Leniency Law)\_{kt}$ + $\delta$$X\_{ikt}$ + $\theta$$\_t$ + $\gamma$$\_i$ +$\epsilon$$\_{it}$ *(1)*

where $i$, $k$, and $t$ index firms, countries, and years respectively. $X\_{ikt}$ is a vector of the different firm, country, and industry control, while $\gamma$ and $\theta$ are firm and year fixed effects.

The variable of interest here is $(Leniency Law)\_{kt}$. [Dasgupta, 2019, p. 2597](https://academic.oup.com/rfs/article/32/7/2587/5079300) documented that this variable equals 0 before the passage of the leniency law in country $k$, and 1 afterward. They follow a standard difference-in-difference approach with staggered implementation of laws ([Bertrand, 2003](https://www.jstor.org/stable/10.1086/376950?seq=1#metadata_info_tab_contents)) to construct the treatment and control group.

**B. Author's description:**

[Dasgupta, 2019, p. 2597](https://academic.oup.com/rfs/article/32/7/2587/5079300) said that

> 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**.

**C. Discussion:**

1. Whether the control group includes firms in countries that implemented this law at some **earlier point in time**? My understanding is: for example, from [Table 2, Dasgupta, 2019, p. 2599](https://academic.oup.com/rfs/article/32/7/2587/5079300), Korea passed the laws in 1997, therefore, the control group including all firms in all countries not implementing this law from 1995 to 2002, including the US.
   From my point of view, it should be the case because based on this identification, firms in the US will not be affected by the laws passed in Korea based on this identification strategy so these US firms from 1995-1996 and 1998-2002 can be control observations for Korea. But if this is the case, **why 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.
2. Does it mean that only $(Leniency Law)\_{kt}$ of the treatment group after the laws being passed received a value 1? Other than that, the control group and the treatment group before the laws being passed receive a value of 0?
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?
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? (meaning no duplicate control observation).
5. Above is how he grouped the control and treatment based on his identification, but on page 2600, [Dasgupta, 2019, section 3.1](https://academic.oup.com/rfs/article/32/7/2587/5079300) 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?

**Tags:** econometrics, international-economics, definition, paneldata, corporate-finance

---

## 2 answers

### Answer by username_2 on 2021-05-18. Score: 4 | Accepted answer

> 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](https://www.nber.org/system/files/working_papers/w25018/w25018.pdf) 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.

> **Comments:**
> **username_0**: thank you for your comprehensive answer, I have a couple of questions as below, much appreciate if you could have a look on that. Regarding the first answer, so, if US enforced the law in 1993, whether US firms from 1995 to2002 is included in the control sample (or counterfactual sample) for Korean ? Even you bold the sentence is the answer 4 but I want to cross check one more time  
> **username_0**: Secondly, I did not get the idea of this sentence, "If we also observed firms i over time within Malaysian industries, then all firms should be coded 0 in every firm-year period.", could you help you to clarify it please? I am wondering why you bring the "industry" up here.  
> **username_0**: Thirdly, I may misunderstand you in this point:"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 research used the window [-2;+5] for their Difference-in-Difference setting. So if Australia adopts leniency in 2005, Germain only be the counterfactual if it passed the laws before 2003. The sample period in section B in my question is [-2;+5] - two years before and five years after the event day. Please correct me if I fall into a fallacy.  
> **username_0**: Fourthly, whether "industry-by-year effects" is running firm fixed effect along with the interaction of year*industry fixed effect, please?  
> **username_2**: (1) Yes. Previously treated countries may serve as counterfactuals for countries treated later. See the paper I referenced which is a detailed treatise on the two-way fixed effects estimator. (2) The fake data frame I provided did not include any data at the $i$-level. The only point I was making is that if you observe firms *within* Malaysia then the treatment dummy is coded 0 for all firms in every single time period.  
> **username_2**: (3) I am under the impression that the firm level data is between 1990–2012. But note that in 'event study' frameworks, we may only consider a finite number of leads and lags around the event date. Suppose Korea's first adoption year is 2000 and Germany's is 2003. The period before exposure (e.g., $-1$) is the year 1999 in Korea and 2002 in Germany. Remember these are relative period effects around the event. Review my answer [here](https://stats.stackexchange.com/questions/523032/testing-parallel-trends-in-staggered-difference-in-differences/523361#523361) for a thorough discussion of this.  
> **username_2**: (4) Yes. The industry * year fixed effects may be included with the firm fixed effects. The authors estimate, separately, industry * year fixed effects *and* region * year fixed effects, both of which were estimated with the firm fixed effects. I believe this was included in Table 3 of Dasgupta's work.  
> **username_2**: Let's jump into a chat if anything else is unclear!  
> **username_0**: Hi @Thomas Bilach, could you please help me to start a chat room, I do not have enough 100 reputations to initiate one chat room(https://stackoverflow.com/help/privileges/chat-rooms#:~:text=How%20do%20I%20create%20a,bottom%20of%20the%20Rooms%20page.). Thank you.  
> **username_2**: Let's see if this works: Enter [here](https://chat.stackexchange.com/rooms/info/124582/difference-in-differences-methods?tab=access.).  

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### Answer by username_1 on 2021-05-12. Score: 1

I think the terms "treatment group" and "control group" are at best a loose analogy in an econometric model with two way fixed effects and staggered adoption of the treatment group. In short, I think you are right to be skeptical of the use of the terms "treatment group" and "control group" here. I think the authors are using them in a less way that is less precise or rigorous way to give a general intuition for the identification strategy by way of an analogy to a randomized trial or a non-staggered difference-in-difference.

There is actually a pretty big recent literature about econometric issues that are somewhat related to this, especially in cases where the treatment effect is not uniform. If you search for "difference in differences" on Twitter you can learn a lot about it.

> **Comments:**
> **username_0**: Thank you @Jonathan Borowsky, I thought you went through this paper a little bit already, I much appreciate it if you can give your own idea about my 5 curiosities above. I did search in Twitter but the knowledge seems not to be concentrated and there is no place to ask. Warm regards and thanks  

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