<p>Generalized DID or Staggered DID are DID using staggered treatment events. In <a href="https://onlinelibrary.wiley.com/doi/epdf/10.1111/1756-2171.12299" rel="nofollow noreferrer">Dong,2019</a>'s paper, he uses the framework as below:</p>
<p><span class="math-container">$Margin_{ikjt}$</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$$X_{ikt}$</span> + <span class="math-container">$\theta$$_{jt}$</span> + <span class="math-container">$\gamma$$_i$</span> +<span class="math-container">$\epsilon$$_{ikjt}$</span> <em>(2)</em></p>
<p>where <span class="math-container">$i$</span>, <span class="math-container">$k$</span>, <span class="math-container">$j$</span> , and <span class="math-container">$t$</span> index firms, countries, industry,  and years respectively. <span class="math-container">$X_{ikt}$</span> is a vector of control variables, while <span class="math-container">$\gamma$</span> and <span class="math-container">$\theta$</span> are firm and year fixed effects. The variable of interest here is <span class="math-container">$(Leniency Law)_{kt}$</span>.</p>
<p>This is the equation (2) above from section &quot;Effects on cartel detections&quot; of <a href="https://onlinelibrary.wiley.com/doi/epdf/10.1111/1756-2171.12299" rel="nofollow noreferrer">Dong,2019's paper, page893-894</a>.
How they assign the samples:</p>
<blockquote>
<p>In our estimation, we rely on the staggered nature of the passage of
leniency programsto identify their causal effect on ﬁrm margins. We
follow the standard approach used in theliterature, which relies on
the staggered passage of laws in different geographic regions
(e.g.,the business combination laws across the United States, as in
Bertrand and Mullainathan, 2003).This allows us to compare the change
in the margins of ﬁrms that were affected by the law to
thecontemporaneous change in the margins of the control ﬁrms that were
headquartered in countries that had not yet passed such a law</p>
</blockquote>
<p>This setting is quite similar to the setting in the paper of <a href="https://academic.oup.com/rfs/article/32/7/2587/5079300" rel="nofollow noreferrer">Dasgupta,2019</a> as <a href="https://economics.stackexchange.com/questions/43864/how-to-justify-the-treatment-and-control-groups-for-difference-in-difference-wit">explained thoroughly here</a>. And these two papers using the same sample classification is predictable because they have same author (Alminas Zaldokas)</p>
<p>From staggered DID, as been <a href="https://economics.stackexchange.com/a/44052/33142">well-explained by Thomas</a> the variable <span class="math-container">$(Leniency Law)_{kt}$</span> is</p>
<blockquote>
<p>The binary treatment variable in this more general setting is not the
same variable as in the 'classical' difference-in-differences case.
Suppose a leniency law is espoused by all firms within treated
countries in the year 2000. In this setting you could write this
equation more simply as the interaction between a treatment-control
dummy and a post-treatment indicator equal to 1 after the law goes
into effect in both groups, 0 otherwise. However, once we move away
from this setting and the roll out of treatment is staggered or even
switching 'on' and 'off' over time, then the &quot;post-treatment&quot; variable
is no longer well-defined. To proceed, we must use the 'generalized'
difference-in-differences estimator which defines the product term in
a different way.</p>
</blockquote>
<p><strong>Below is my important critics about the summary statistics of <a href="https://academic.oup.com/rfs/article/32/7/2587/5079300" rel="nofollow noreferrer">Dong, 2019</a>'s paper</strong></p>
<p>From the description above, the observations of control group should be higher than that of treatment group. Apart from that, in their summary statistic (Table 2, page 889)</p>
<p><a href="https://i.sstatic.net/PNTLY.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/PNTLY.png" alt="enter image description here" /></a></p>
<p><em><strong>It is totally not the case because <span class="math-container">$(Leniency Law)_{kt}$</span> only receive the value of 1 when it satisfies both conditions: in the treatment group and during post-treatment period. Therefore, if my logical argument above is correct, there is no way that Leniency law has the mean of 0.614 (more than one half of the concatenating sample of treatment and control group)</strong></em></p>
<p>Update:
Thank to the discussion with @1muflon1, I clarify more about why the observations of control should be higher than that of treatment countries.</p>
<blockquote>
<p>For any country, the observations of control sample must be higher than
that of the treatment sample. This is because, every country always
being treated for 5 years (because this research used the window
<strong>[-2;+2]</strong>;  and during the rest period from 1990 to 2012 (excepting the
treatment years), this country belongs to control sample based on the setting of
generalized DID</p>
</blockquote>

CC BY-SA 4.0
2021-06-01T08:39:47.987Z
Maybe you can write a friendly email to the authors of the paper to ask for further clarification?
tdm
CC BY-SA 4.0
2021-06-01T08:40:39.883Z
@tdm Actually, I wrote a couple of emails but did not receive the answer tho...
Phil Nguyen
CC BY-SA 4.0
2021-06-01T09:24:08.593Z
I am not sure if I follow the argument why would the number of observations in control group need to be more than the ones in the treatment group based on the descriptions above?
1muflon1
CC BY-SA 4.0
2021-06-01T09:30:12.670Z
[1/2] @1muflon1 It is relatively speaking, the whole sample period is from 1990 to 2012. And based on the explanation of Thomas in the link [here](https://economics.stackexchange.com/questions/43864/how-to-justify-the-treatment-and-control-groups-for-difference-in-difference-wit/44052#44052), if Korea implements the law in 2000, so the variables Leniency Laws only receive the value of 1 in the year 2001 and 2002 and receiving the value of 0 in all years from 1990 to 2012 (except 2001 and 2002). The window is [-2;+2].
Phil Nguyen
CC BY-SA 4.0
2021-06-01T09:30:20.323Z
[2/2] Therefore, it seems that the observations in the control group should be higher than those in the treatment group.
Phil Nguyen
CC BY-SA 4.0
2021-06-01T09:36:59.173Z
@BeautifulMindset but the sample has multiple countries in it. You can very well have 1 untreated country which would have always zero (lets for a sake of argument say we have  5 countries (A,B,C, D, E) and 3 years (1,2,3)) So you could  have 1 untreated country with entries 0,0,0 and then 4countries that got treatment at 2 so they will have entries 0, 1, 1; 0, 1, 1; 0,1,1 and 0,1,1 here the mean will be approx 0.533, unless you have examined whole sample how can you make argument based on existence of 1 untreated country?
1muflon1
CC BY-SA 4.0
2021-06-01T09:47:50.003Z
[1/2] @1muflon1 Thank you for your example, which helps me to clarify my idea. First, in your sample, the number of treatment countries is higher than that of control countries while in the research, the numbers of control countries are higher than that of treatment countries.
Phil Nguyen
CC BY-SA 4.0
2021-06-01T09:50:38.240Z
[2/2]Second, your time span is quite short compared to the research's time span, which incidentally biases the result. For example, for any country, the control sample must be higher than that of the treatment variable. This is because, every country always being treatment for 5 years (because this research used the window [-2;+2] and during the rest period from 1990 to 2012 (excepting the treatment years), this country is control based on the setting of generalized DID.
Phil Nguyen
CC BY-SA 4.0
2021-06-01T09:55:49.753Z
@BeautifulMindset thanks for explanation maybe consider putting it in the question itself
1muflon1
CC BY-SA 4.0
2021-06-01T10:23:14.310Z
@1muflon1 Thank you so much, thanks to discussing with you, I know that I need to improve my question to let it be better, I hope my explanation makes sense tho.
Phil Nguyen
CC BY-SA 4.0
2021-06-01T11:21:37.547Z
N is number of observations which is company years. USA is treated in 1993 very early in the period. USA has 22.498 firms amounting to 213.914 observations=company years. Observations are spread across T=23 timeperiods. 3 periods without treatment and 20 with so 87% of the timeduration USA is in treatment.  Assuming the 213.914 observations are randomly distributed across timeperiods amounts to 0.87*213.914= 186.012 treated observations. And 186012/543737 = 34.2% of all observations being treament. And this is just accounting for USA.
Jesper Hybel
CC BY-SA 4.0
2021-06-01T11:29:40.730Z
@JesperHybel yes, but maybe a mistype that "3 periods **with** treatment and 20 **without**". So, it seems that your calculation has things to concern, meaning 87% of US in control rather than treatment. Please correct me if I misunderstand you
Phil Nguyen
CC BY-SA 4.0
2021-06-01T11:34:33.483Z
No not a mistype ... 1990,1991,1992 are years before law is passed in US ...  that is 3 years where (LeniencyLaw)kt = 0 for k=USA.
Jesper Hybel
CC BY-SA 4.0
2021-06-01T11:38:26.850Z
@JesperHybel , so, the author uses the window **[-2;+2]**. it means that in US case, it enforces the law in 1993, it means that only the years 1993,1994,1995 are treatment, the rest period from 1991,1992, 1996->2012, the LeniencyLaw variable equal to 0 (because 1991->1992 are pre-treatment period and 1996->2012 belong to control sample). Hope I understand you correctly.
Phil Nguyen
CC BY-SA 4.0
2021-06-01T11:46:06.180Z
No you are confusing yourself. You insist on talking about treatment and control. The window is completely irrelevant. The mean of Leniency Law is the mean of a dummy variable. The definition of this dummy variable is that it is 1 if company i at time t is in country k and country k has passed Leniency Law at that time, otherwise it is 0.
Jesper Hybel
CC BY-SA 4.0
2021-06-01T11:47:49.473Z
If I use Table 1 for the entire sample and assume that companies are randomly distributed across time that implies 51% observations on Leniency Law should be 1 so 61% does not seem out of order in my opinion.
Jesper Hybel
CC BY-SA 4.0
2021-06-01T11:54:40.860Z
@JesperHybel . Thank you for your clarification. However, I think, every company has a Leniency Law variable and it varies based on the treatment periods. For example, a company A in the US will have Leniency Law=1 in 1993,1994,1995 and have Leniency Law=0 in 1991,1992, 1996->2012 from my point of view. Therefore, mean of Leniency Law of a random firm A in the US equalling to **3/23** . The variable Leniency Laws here is treat*post in standard DID. Please correct me if I fall into any fallacy then.
Phil Nguyen
CC BY-SA 4.0
2021-06-01T11:57:51.277Z
Why would LeniencyLaw stop being 1 in1996 and continue to be 0 until 2012 (for the case of USA)?
Jesper Hybel
CC BY-SA 4.0
2021-06-01T12:01:41.480Z
@JesperHybel , yesss, it is the tricky thing of staggered DID to me. It is because the window is [-2;+2], so Leniency Law only equals 1 in 1993 to 1995, afterward, the US are controls for other countries, which means the US belongs to control groups from 1996, meaning Leniency Law equals 0 in such a case.
Phil Nguyen
CC BY-SA 4.0
2021-06-13T07:39:12.583Z
@JesperHybel I just want to come back and say that you are right in your arguments. Thank you so much.
Phil Nguyen
<econometrics><international-economics><paneldata>
A doubt on summary statistic of post*treat variable of a paper using DID (Dong,2019)
Phil Nguyen
# A doubt on summary statistic of post*treat variable of a paper using DID (Dong,2019)

(Asked by: username_0 on 2021-06-01)

Generalized DID or Staggered DID are DID using staggered treatment events. In [Dong,2019](https://onlinelibrary.wiley.com/doi/epdf/10.1111/1756-2171.12299)'s paper, he uses the framework as below:

$Margin\_{ikjt}$ = $\alpha$ + $\beta$ $(Leniency Law)\_{kt}$ + $\delta$$X\_{ikt}$ + $\theta$$\_{jt}$ + $\gamma$$\_i$ +$\epsilon$$\_{ikjt}$ *(2)*

where $i$, $k$, $j$ , and $t$ index firms, countries, industry, and years respectively. $X\_{ikt}$ is a vector of control variables, while $\gamma$ and $\theta$ are firm and year fixed effects. The variable of interest here is $(Leniency Law)\_{kt}$.

This is the equation (2) above from section "Effects on cartel detections" of [Dong,2019's paper, page893-894](https://onlinelibrary.wiley.com/doi/epdf/10.1111/1756-2171.12299).
How they assign the samples:

> In our estimation, we rely on the staggered nature of the passage of
> leniency programsto identify their causal effect on ﬁrm margins. We
> follow the standard approach used in theliterature, which relies on
> the staggered passage of laws in different geographic regions
> (e.g.,the business combination laws across the United States, as in
> Bertrand and Mullainathan, 2003).This allows us to compare the change
> in the margins of ﬁrms that were affected by the law to
> thecontemporaneous change in the margins of the control ﬁrms that were
> headquartered in countries that had not yet passed such a law

This setting is quite similar to the setting in the paper of [Dasgupta,2019](https://academic.oup.com/rfs/article/32/7/2587/5079300) as [explained thoroughly here](https://economics.stackexchange.com/questions/43864/how-to-justify-the-treatment-and-control-groups-for-difference-in-difference-wit). And these two papers using the same sample classification is predictable because they have same author (Alminas Zaldokas)

From staggered DID, as been [well-explained by Thomas](https://economics.stackexchange.com/a/44052/33142) the variable $(Leniency Law)\_{kt}$ is

> 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.

**Below is my important critics about the summary statistics of [Dong, 2019](https://academic.oup.com/rfs/article/32/7/2587/5079300)'s paper**

From the description above, the observations of control group should be higher than that of treatment group. Apart from that, in their summary statistic (Table 2, page 889)

[![enter image description here](https://i.sstatic.net/PNTLY.png)](https://i.sstatic.net/PNTLY.png)

***It is totally not the case because $(Leniency Law)\_{kt}$ only receive the value of 1 when it satisfies both conditions: in the treatment group and during post-treatment period. Therefore, if my logical argument above is correct, there is no way that Leniency law has the mean of 0.614 (more than one half of the concatenating sample of treatment and control group)***

Update:
Thank to the discussion with @username_2, I clarify more about why the observations of control should be higher than that of treatment countries.

> For any country, the observations of control sample must be higher than
> that of the treatment sample. This is because, every country always
> being treated for 5 years (because this research used the window
> **[-2;+2]**; and during the rest period from 1990 to 2012 (excepting the
> treatment years), this country belongs to control sample based on the setting of
> generalized DID

**Tags:** econometrics, international-economics, paneldata

> **Comments:**
> **username_1**: Maybe you can write a friendly email to the authors of the paper to ask for further clarification?  
> **username_0**: @username_1 Actually, I wrote a couple of emails but did not receive the answer tho...  
> **username_2**: I am not sure if I follow the argument why would the number of observations in control group need to be more than the ones in the treatment group based on the descriptions above?  
> **username_0**: [1/2] @username_2 It is relatively speaking, the whole sample period is from 1990 to 2012. And based on the explanation of Thomas in the link [here](https://economics.stackexchange.com/questions/43864/how-to-justify-the-treatment-and-control-groups-for-difference-in-difference-wit/44052#44052), if Korea implements the law in 2000, so the variables Leniency Laws only receive the value of 1 in the year 2001 and 2002 and receiving the value of 0 in all years from 1990 to 2012 (except 2001 and 2002). The window is [-2;+2].  
> **username_0**: [2/2] Therefore, it seems that the observations in the control group should be higher than those in the treatment group.  
> **username_2**: @BeautifulMindset but the sample has multiple countries in it. You can very well have 1 untreated country which would have always zero (lets for a sake of argument say we have  5 countries (A,B,C, D, E) and 3 years (1,2,3)) So you could  have 1 untreated country with entries 0,0,0 and then 4countries that got treatment at 2 so they will have entries 0, 1, 1; 0, 1, 1; 0,1,1 and 0,1,1 here the mean will be approx 0.533, unless you have examined whole sample how can you make argument based on existence of 1 untreated country?  
> **username_0**: [1/2] @username_2 Thank you for your example, which helps me to clarify my idea. First, in your sample, the number of treatment countries is higher than that of control countries while in the research, the numbers of control countries are higher than that of treatment countries.  
> **username_0**: [2/2]Second, your time span is quite short compared to the research's time span, which incidentally biases the result. For example, for any country, the control sample must be higher than that of the treatment variable. This is because, every country always being treatment for 5 years (because this research used the window [-2;+2] and during the rest period from 1990 to 2012 (excepting the treatment years), this country is control based on the setting of generalized DID.  
> **username_2**: @BeautifulMindset thanks for explanation maybe consider putting it in the question itself  
> **username_0**: @username_2 Thank you so much, thanks to discussing with you, I know that I need to improve my question to let it be better, I hope my explanation makes sense tho.  
> **username_3**: N is number of observations which is company years. USA is treated in 1993 very early in the period. USA has 22.498 firms amounting to 213.914 observations=company years. Observations are spread across T=23 timeperiods. 3 periods without treatment and 20 with so 87% of the timeduration USA is in treatment.  Assuming the 213.914 observations are randomly distributed across timeperiods amounts to 0.87*213.914= 186.012 treated observations. And 186012/543737 = 34.2% of all observations being treament. And this is just accounting for USA.  
> **username_0**: @username_3 yes, but maybe a mistype that "3 periods **with** treatment and 20 **without**". So, it seems that your calculation has things to concern, meaning 87% of US in control rather than treatment. Please correct me if I misunderstand you  
> **username_3**: No not a mistype ... 1990,1991,1992 are years before law is passed in US ...  that is 3 years where (LeniencyLaw)kt = 0 for k=USA.  
> **username_0**: @username_3 , so, the author uses the window **[-2;+2]**. it means that in US case, it enforces the law in 1993, it means that only the years 1993,1994,1995 are treatment, the rest period from 1991,1992, 1996->2012, the LeniencyLaw variable equal to 0 (because 1991->1992 are pre-treatment period and 1996->2012 belong to control sample). Hope I understand you correctly.  
> **username_3**: No you are confusing yourself. You insist on talking about treatment and control. The window is completely irrelevant. The mean of Leniency Law is the mean of a dummy variable. The definition of this dummy variable is that it is 1 if company i at time t is in country k and country k has passed Leniency Law at that time, otherwise it is 0.  
> **username_3**: If I use Table 1 for the entire sample and assume that companies are randomly distributed across time that implies 51% observations on Leniency Law should be 1 so 61% does not seem out of order in my opinion.  
> **username_0**: @username_3 . Thank you for your clarification. However, I think, every company has a Leniency Law variable and it varies based on the treatment periods. For example, a company A in the US will have Leniency Law=1 in 1993,1994,1995 and have Leniency Law=0 in 1991,1992, 1996->2012 from my point of view. Therefore, mean of Leniency Law of a random firm A in the US equalling to **3/23** . The variable Leniency Laws here is treat*post in standard DID. Please correct me if I fall into any fallacy then.  
> **username_3**: Why would LeniencyLaw stop being 1 in1996 and continue to be 0 until 2012 (for the case of USA)?  
> **username_0**: @username_3 , yesss, it is the tricky thing of staggered DID to me. It is because the window is [-2;+2], so Leniency Law only equals 1 in 1993 to 1995, afterward, the US are controls for other countries, which means the US belongs to control groups from 1996, meaning Leniency Law equals 0 in such a case.  
> **username_0**: @username_3 I just want to come back and say that you are right in your arguments. Thank you so much.  

---

## 0 answers

