# Using Multiple Pre-treatment Periods to Improve
Difference-in-Differences and Staggered Adoption Designs††thanks:  The methods
proposed in this article can be implemented via the open-source statistical
software R package DIDdesign available at https://github.com/naoki-
egami/DIDdesign. We are grateful to Edmund Malesky, Cuong Viet Nguyen, and Anh
Tran for providing us with data and answering our questions. We thank Adam
Glynn, Chad Hazlett, Shiro Kuriwaki, Ian Lundberg, John Marshall, Xiang Zhou,
and participants of the 2019 Summer Meetings of the Political Methodology
Society and the 2019 American Political Science Association Annual Conference
for helpful comments and discussions. We also thank the editor and our two
anonymous reviewers for providing us with valuable comments.

Naoki Egami  Assistant Professor, Department of Political Science, Columbia
University, New York NY 10027. Email<EMAIL_ADDRESS>URL:
https://naokiegami.com.  Soichiro Yamauchi  PhD Candidate, Department of
Government, Harvard University, Cambridge MA 02138. Email:
<EMAIL_ADDRESS>URL: https://soichiroy.github.io

( This version: February 3, 2022
First draft: December 6, 2019 )

###### Abstract

While a difference-in-differences (DID) design was originally developed with
one pre- and one post-treatment period, data from additional pre-treatment
periods are often available. How can researchers improve the DID design with
such multiple pre-treatment periods under what conditions? We first use
potential outcomes to clarify three benefits of multiple pre-treatment
periods: (1) assessing the parallel trends assumption, (2) improving
estimation accuracy, and (3) allowing for a more flexible parallel trends
assumption. We then propose a new estimator, double DID, which combines all
the benefits through the generalized method of moments and contains the two-
way fixed effects regression as a special case. We show that the double DID
requires a weaker assumption about outcome trends and is more efficient than
existing DID estimators. We also generalize the double DID to the staggered
adoption design where different units can receive the treatment in different
time periods. We illustrate the proposed method with two empirical
applications, covering both the basic DID and staggered adoption designs. We
offer an open-source R package that implements the proposed methodologies.

##  1 Introduction

Over the last few decades, social scientists have developed and applied
various approaches to make credible causal inference from observational data.
One of the most popular is a difference-in-differences (DID) design [9, 3].
When the outcome trend of the control group would have been the same as the
trend of the outcome in the treatment group in the absence of the treatment
(known as the parallel trends assumption), the DID design enables scholars to
estimate causal effects even in the presence of time-invariant unmeasured
confounding [1]. In its most basic form, we compare treatment and control
groups over two time periods — one before and the other after the treatment
assignment.

In practice, it is common to apply the DID method with additional pre-
treatment periods.111In our literature review of American Political Science
Review and American Journal of Political Science between 2015 and 2019, we
found that about 63% of the papers that use the DID design have more than one
pre-treatment period. See Appendix A for details about our literature review.
However, in contrast to the basic two-time-period case, there are a number of
different ways to analyze the DID design with multiple pre-treatment periods.
One popular approach is to apply the two-way fixed effects regression to the
entire time periods and supplement it with alternative model specifications by
including time-trends or leads of the treatment variable to assess possible
violations of the parallel trends assumption. Another is to stick with the
two-time-period DID and limit the use of additional pre-treatment periods only
to the assessment of pre-treatment trends.222For each approach, we provide
examples in Appendix A. This variation of approaches raises an important
practical question: how should analysts incorporate multiple pre-treatment
periods into the DID design, and under what assumptions? In Section 2, we
begin by examining three benefits of multiple pre-treatment periods using
potential outcomes [24]: (1) assessing the parallel trends assumption, (2)
improving estimation accuracy, and (3) allowing for a more flexible parallel
trends assumption. While these benefits have been discussed in the literature,
we revisit them to clarify that each benefit requires different assumptions
and estimators. As a result, in practice, researchers tend to enjoy only a
subset of the three benefits they can exploit from multiple pre-treatment
periods. While our literature review finds that more than 90% of papers based
on the DID design enjoy at least one of the three benefits, we also find that
only 20% of the papers enjoy all three benefits.

Our main contribution is to propose a new, simple estimator that achieves all
three benefits together. We use the generalized method of moments (GMM)
framework [18] to develop the double difference-in-differences (double DID).
At its core, we combine two popular DID estimators: the standard DID
estimator, which relies on the canonical parallel-trends assumptions, and the
sequential DID estimator [28, 32, e.g.,], which only requires that the change
in the trends is the same across treatment and control groups (what we call
the parallel trends-in-trends assumption). While each estimator itself is not
new, the new combination of the two estimators via the GMM allows us to
optimally exploit the three benefits of multiple pre-treatment periods.

The proposed double DID approach makes several key methodological
contributions. First, we show that the proposed method achieves better
theoretical properties than widely-used DID estimators that constitute the
double DID. Compared to the standard DID estimator and the two-way fixed
effects regression, the double DID has smaller standard errors (i.e., more
efficient) and is unbiased under a weaker assumption. While the former
estimators require the parallel trends assumption, the double DID only
requires the parallel trends-in-trends assumption. The double DID also
improves upon the sequential DID estimator, which is inefficient when the
parallel trends assumption holds. By using the GMM theory, we show that the
double DID is more efficient than the sequential DID when the parallel trends
assumption holds. Therefore, our proposed GMM approach enables methodological
improvement both in terms of identification and estimation accuracy.

Second, and most importantly in practice, the double DID blends all the three
benefits of multiple pre-treatment periods within a single framework.
Therefore, instead of using different estimators for enjoying each benefit as
required in existing methods, researchers can use the double DID approach to
exploit all the benefits. Given that only 20% of papers based on the DID
design currently enjoy all the three benefits, our proposed unified approach
to optimally exploit all the three benefits of multiple pre-treatment periods
is essential in practice.

We also propose three extensions of our double DID estimator. First, we
develop the double DID regression, which can incorporate pre-treatment
observed covariates to make the DID design more robust and efficient (Section
3.3.1). Second, we allow for any number of pre\- and post-treatment periods
(Section 3.3.2). While the parallel trends-in-trends assumption can allow for
time-varying unmeasured confounders that linearly change over time, we show
how to further relax the assumption by accounting for even more flexible forms
of time-varying unmeasured confounding when we have more pre-treatment
periods. Because our proposed methods allow for any number of post-treatment
periods, researchers can also estimate not only short-term causal effects but
also longer-term causal effects. Finally, we generalize our double DID
estimator to the staggered adoption design where different units can receive
the treatment in different time periods (Section 4). This design is
increasingly more popular in political science and social sciences [7, 5, 31,
e.g.,].

We offer a companion R package DIDdesign that implements the proposed methods.
We illustrate our proposed methods through two empirical applications. In
Section 3.4, we revisit [30], which study how the abolition of elected
councils affects local public services. This serves as an example of the basic
DID design where treatment assignment happens only once. In Appendix H.2, we
reanalyze [34], which examines the effect of granting collective bargaining
rights to teacher’s unions on educational expenditures and teacher’s salaries.
This is an example of the staggered adoption design.

##### Related Literature.

This paper builds on the large literature of time-series cross-sectional data.
Generalizing the well-known case of two periods and two groups [1, e.g.,],
recent papers use potential outcomes to unpack the nonparametric connection
between the DID and two-way fixed effects regression estimators, thereby
proposing extensions to relax strong parametric and causal assumptions [36,
23, 11, 5, 16, 22, e.g.,]. Our paper also uses potential outcomes to clarify
nonparametric foundations on the use of multiple pre-treatment periods. The
key difference is that, while this recent literature mainly considers
identification under the parallel trends assumption, we study both estimation
accuracy and identification under more flexible assumptions of trends. We do
so both in the basic DID setup and in the staggered adoption design.

Another class of popular methods is the synthetic control method [2] and their
recent extensions [40, 7, 35, e.g.,] that estimate a weighted average of
control units to approximate a treated unit. As carefully noted in those
papers, such methodologies require long pre-treatment periods to accurately
estimate a pre-treatment trajectory of the treated unit [2]; for example, [40]
recommends collecting more than ten pre-treatment periods. In contrast, the
proposed double DID can be applied as long as there is more than one pre-
treatment period, and is better suited when there are a small to moderate
number of pre-treatment periods.333In our literature review, we found that
most DID applications have less than 10 pre-treatment periods, and the median
number of pre-treatment periods is $3.5$. See Appendix A for more details.
However, we also show in Appendix H.2 that the double DID can achieve
performance comparable to variants of synthetic control methods even when
there are a large number of pre-treatment periods. We offer additional
discussions about relationships between our proposed approach and synthetic
control methods in Appendix B.

##  2 Three Benefits of Multiple Pre-treatment Periods

The difference-in-differences (DID) design is one of the most widely used
methods to make causal inference from observational studies. The basic DID
design consists of treatment and control groups measured at two time periods,
before and after the treatment assignment. While the basic DID design only
requires data from one post- and one pre-treatment period, additional pre-
treatment periods are often available. Unfortunately, however, assumptions
behind different uses of multiple pre-treatment periods have often remained
unstated.

In this section, we use potential outcomes to discuss three well-known
practical benefits of multiple pre-treatment periods: (1) assessing the
parallel trends assumption, (2) improving estimation accuracy, and (3)
allowing for a more flexible parallel trends assumption. This section serves
as a methodological foundation for developing a new approach in Sections 3 and
4.

As our running example, we focus on a study of how the abolition of elected
councils affects local public services. [30] use the DID design to examine the
effect of recentralization efforts in Vietnam. The abolition of elected
councils, the main treatment of interest, was implemented in 2009 in about 12%
of all the communes, which are the smallest administrative units that the
paper considers. For each commune, a variety of outcomes related to public
services, such as the quality of infrastructure, were measured in 2006, 2008,
and 2010\. With this data, [30] aim to estimate the causal effect of
abolishing elected councils on various measures of local public services.

###  2.1 Setup

To begin with, let $D_{it}$ denote the binary treatment for unit $i$ in time
period $t$ so that $D_{it}=1$ if the unit is treated in time period $t$, and
$D_{it}=0$ otherwise. In this section, we consider two pre-treatment time
periods $t\in\\{0,1\\}$ and one post-treatment period $t=2$. We choose this
setup here because it is sufficient for examining benefits of multiple pre-
treatment periods, but we also generalize our methods to an arbitrary number
of pre\- and post- treatment periods (Section 3.3.2), and to the staggered
adoption design (Section 4). In our example, two pre-treatment periods are
2006 and 2008, and one post-treatment period is 2010\. Thus, the treatment
group receives the treatment only at time $t=2$; $D_{i0}=D_{i1}=0$ and
$D_{i2}=1$, whereas units in the control group never gets treated
$D_{i0}=D_{i1}=D_{i2}=0$. We refer to the treatment group as $G_{i}=1$ and the
control group as $G_{i}=0.$ Outcome $Y_{it}$ is measured at time
$t\in\\{0,1,2\\}.$ In addition to panel data where the same units are measured
over time, the DID design accommodates repeated cross-sectional data, in which
different communes are sampled at three time periods.

To define causal effects, we rely on the potential outcomes framework [24].
For each time period, $Y_{it}(1)$ represents the quality of infrastructure
that commune $i$ would achieve in time period $t$ if commune $i$ had abolished
elected councils. $Y_{it}(0)$ is similarly defined. For an individual commune,
the causal effect of abolishing elected councils on the quality of
infrastructure in time period $t$ is $Y_{it}(1)-Y_{it}(0)$. As the treatment
is assigned in the second time period, we are interested in estimating a
causal effect at time $t=2$, and a causal effect of interest is formally
defined as $Y_{i2}(1)-Y_{i2}(0).$

In the DID design, we are interested in estimating the average treatment
effect for treated units (ATT) [3]:

$\tau=\mathbb{E}[Y_{i2}(1)-Y_{i2}(0)\mid G_{i}=1],$ (1)

where the expectation is over units in the treatment group $G_{i}=1$.

#### DID with One Pre-Treatment Period

Before we discuss benefits of multiple pre-treatment periods from Section 2.2,
we briefly review the DID with one pre-treatment period to fix ideas for
settings with multiple pre-treatment periods.

In the basic DID design, researchers can identify the ATT based on the widely-
used assumption of parallel trends — if the treatment group had not received
the treatment in the second period, its outcome trend would have been the same
as the trend of the outcome in the control group. [3].

######  Assumption 1 (Parallel Trends).

$\mathbb{E}[Y_{i2}(0)\mid G_{i}=1]-\mathbb{E}[Y_{i1}(0)\mid G_{i}=1]\ =\
\mathbb{E}[Y_{i2}(0)\mid G_{i}=0]-\mathbb{E}[Y_{i1}(0)\mid G_{i}=0].$ (2)

The left-hand side of equation (2) is the trend in outcomes for the treatment
group $G_{i}=1$, and the right is the one for the control group $G_{i}=0$.
Under the parallel trends assumption, we estimate the ATT via the difference-
in-differences estimator.

$\widehat{\tau}_{\texttt{DID}}=\left(\frac{\sum_{i\colon
G_{i}=1}Y_{i2}}{n_{12}}-\frac{\sum_{i\colon
G_{i}=1}Y_{i1}}{n_{11}}\right)-\left(\frac{\sum_{i\colon
G_{i}=0}Y_{i2}}{n_{02}}-\frac{\sum_{i\colon G_{i}=0}Y_{i1}}{n_{01}}\right),$
(3)

where $n_{1t}$ and $n_{0t}$ are the numbers of units in the treatment and
control groups at time $t\in\\{1,2\\}$, respectively.

When we analyze panel data, we can compute $\widehat{\tau}_{\texttt{DID}}$
nonparametrically via a linear regression with unit and time fixed effects.
This numerical equivalence in the two-time-period case is often used to
justify the two-way fixed effects regression as the DID design [3]. We discuss
additional results on nonparametric equivalence between a regression estimator
and the DID estimator in Appendix C.1.

###  2.2 Benefit 1: Assessing Parallel Trends Assumption

We now consider how researchers can exploit multiple pre-treatment periods,
while clarifying necessary underlying assumptions.

The first and the most common use of multiple pre-treatment periods is to
assess the identification assumption of parallel trends. As the validity of
the DID design rests on this assumption, it is critical to evaluate its
plausibility in any application. However, the parallel trends assumption
itself involves counterfactual outcomes, and thus analysts cannot empirically
test it directly. Instead, we often investigate whether trends for treatment
and control groups are parallel in pre-treatment periods as a placebo test
[3].

Specifically, researchers often estimate the DID for the pre-treatment
periods:

$\left(\frac{\sum_{i\colon G_{i}=1}Y_{i1}}{n_{11}}-\frac{\sum_{i\colon
G_{i}=1}Y_{i0}}{n_{10}}\right)-\left(\frac{\sum_{i\colon
G_{i}=0}Y_{i1}}{n_{01}}-\frac{\sum_{i\colon G_{i}=0}Y_{i0}}{n_{00}}\right).$
(4)

We then check whether the DID estimate on pre-treatment periods is
statistically distinguishable from zero. For example, we can apply the DID
estimator to 2006 and 2008 as if 2008 were the post-treatment period, and
assess whether the estimate would be close to zero. In Figure 1, a DID
estimate on the pre-treatment periods would be close to zero for the left
panel, while it would be negative for the right panel where two groups have
different pre-treatment trends. In Appendix C.4, we show that a robustness
check with leads effects [3], which incorporates leads of the treatment
variable into the two-way fixed effects regression and checks whether their
coefficients are zero, is equivalent to this DID on the pre-treatment periods.

Figure 1: Parallel Pre-treatment Trends (left) and Non-Parallel Pre-treatment
Trends (right).

The basic idea behind this test is that if trends are parallel from 2006 to
2008, it is more likely that the parallel trends assumption holds for 2008 and
2010. Hence, instead of considering parallel trends only from 2008 to 2010,
the test evaluates the two related parallel trends together. By doing so, this
popular test tries to make the DID design falsifiable.

Importantly, this approach does not test the parallel trends assumption itself
(Assumption 1), which is untestable due to counterfactual outcomes. Instead,
it tests the extended parallel trends assumption — the parallel trends hold
for pre-treatment periods, from $t=0$ to $t=1$, as well as from a pre-
treatment period $t=1$ to a post-treatment period $t=2$:

######  Assumption 2 (Extended Parallel Trends).

$\begin{cases}\mathbb{E}[Y_{i2}(0)\mid G_{i}=1]-\mathbb{E}[Y_{i1}(0)\mid
G_{i}=1]=\mathbb{E}[Y_{i2}(0)\mid G_{i}=0]-\mathbb{E}[Y_{i1}(0)\mid
G_{i}=0]\\\ \mathbb{E}[Y_{i1}(0)\mid G_{i}=1]-\mathbb{E}[Y_{i0}(0)\mid
G_{i}=1]=\mathbb{E}[Y_{i1}(0)\mid G_{i}=0]-\mathbb{E}[Y_{i0}(0)\mid
G_{i}=0]\end{cases}$ (5)

The first line of the extended parallel trends assumption is the same as the
standard parallel trends assumption, and the second line is the parallel
trends for pre-treatment periods. Because outcome trends are observable in
pre-treatment periods, the test of pre-treatment trends (equation (4))
directly tests this assumption.

It is important to emphasize that, even if we find the DID estimate on pre-
treatment periods is close to zero, we cannot confirm the extended parallel
trends assumption (Assumption 2) or the parallel trends assumption (Assumption
1). This is because it is still possible that trends between $t=1$ (pre-
treatment) and $t=2$ (post-treatment) are not parallel. Therefore, it is
always important to substantively justify the parallel trends assumption in
addition to using this statistical test based on pre-treatment trends.

###  2.3 Benefit 2: Improving Estimation Accuracy

As we discussed above, many existing DID studies that utilize the test of pre-
treatment trends can be viewed as the DID design with the extended parallel
trends assumption. However, this extended parallel trends assumption is often
made implicitly, and thus, it is used only for assessing the parallel trends
assumption. Fortunately, if the extended parallel trends assumption holds, we
can also estimate the ATT with higher accuracy, resulting in smaller standard
errors.

This additional benefit becomes clear by simply restating the extended
parallel trends assumption as follows.

$\begin{cases}\mathbb{E}[Y_{i2}(0)\mid G_{i}=1]-\mathbb{E}[Y_{i1}(0)\mid
G_{i}=1]=\mathbb{E}[Y_{i2}(0)\mid G_{i}=0]-\mathbb{E}[Y_{i1}(0)\mid
G_{i}=0]\\\ \mathbb{E}[Y_{i2}(0)\mid G_{i}=1]-\mathbb{E}[Y_{i0}(0)\mid
G_{i}=1]=\mathbb{E}[Y_{i2}(0)\mid G_{i}=0]-\mathbb{E}[Y_{i0}(0)\mid
G_{i}=0].\end{cases}$ (6)

Under the extended parallel trends assumption, there are two natural DID
estimators for the ATT.

$\displaystyle\widehat{\tau}_{\texttt{DID}}$
$\displaystyle=\left(\frac{\sum_{i\colon
G_{i}=1}Y_{i2}}{n_{12}}-\frac{\sum_{i\colon
G_{i}=1}Y_{i1}}{n_{11}}\right)-\left(\frac{\sum_{i\colon
G_{i}=0}Y_{i2}}{n_{02}}-\frac{\sum_{i\colon G_{i}=0}Y_{i1}}{n_{01}}\right),$
$\displaystyle\widehat{\tau}_{\texttt{DID(2,0)}}$
$\displaystyle=\left(\frac{\sum_{i\colon
G_{i}=1}Y_{i2}}{n_{12}}-\frac{\sum_{i\colon
G_{i}=1}Y_{i0}}{n_{10}}\right)-\left(\frac{\sum_{i\colon
G_{i}=0}Y_{i2}}{n_{02}}-\frac{\sum_{i\colon G_{i}=0}Y_{i0}}{n_{00}}\right).$
(7)

Under the extended parallel trends assumption, both estimators are unbiased
and consistent for the ATT. Thus, we can increase estimation accuracy by
combining the two estimators, for example, simply averaging them.

$\widehat{\tau}_{\texttt{e-DID}}=\frac{1}{2}\widehat{\tau}_{\texttt{DID}}+\frac{1}{2}\widehat{\tau}_{\texttt{DID(2,0)}}.$
(8)

Intuitively, this extended DID estimator is more efficient because we have
more observations to estimate counterfactual outcomes for the treatment group
$\mathbb{E}[Y_{i2}(0)\mid G_{i}=1]$.

In the panel data settings, we show that this extended DID estimator
$\widehat{\tau}_{\texttt{e-DID}}$ is equivalent to the two-way fixed effects
estimator fitted to the three periods $t\in\\{0,1,2\\}$.

$Y_{it}\sim\alpha_{i}+\delta_{t}+\beta D_{it},$ (9)

where $\alpha_{i}$ is a unit fixed effect, $\delta_{t}$ is a time fixed
effect, and a coefficient of the treatment variable $\beta$ is numerically
equal to $\widehat{\tau}_{\texttt{e-DID}}$. We also present more general
results about nonparametric relationships between the extended DID and the
two-way fixed effects estimator in Appendix C.2.

###  2.4 Benefit 3: Allowing For A More Flexible Parallel Trends Assumption

In this section, we consider scenarios in which the extended parallel trends
assumption may not be plausible. Multiple pre-treatment periods are also
useful in accounting for some deviation from the parallel trends assumption.
We discuss a popular generalization of the difference-in-differences
estimator, a _sequential_ DID estimator, which removes bias due to certain
violations of the parallel trends assumption [28, 32, e.g.,]. We clarify an
assumption behind this simple method and relate it to the parallel trends
assumption.

To introduce the sequential DID estimator, we begin with the extended parallel
trends assumption. As we described in Section 2.2, when the extended parallel
trends assumption holds, a DID estimator applied to pre-treatment periods
$t=0$ and $t=1$ should be zero in expectation. In contrast, when trends of
treatment and control groups are not parallel, a DID estimate on pre-treatment
periods would be non-zero. The sequential DID estimator uses this DID estimate
from pre-treatment periods to adjust for bias in the standard DID estimator.
In particular, it subtracts the DID estimator on pre-treatment periods from
the standard DID estimator that uses pre- and post-treatment periods $t=1$ and
$t=2.$

$\displaystyle\widehat{\tau}_{\texttt{s-DID}}$
$\displaystyle=\left\\{\left(\frac{\sum_{i\colon
G_{i}=1}Y_{i2}}{n_{12}}-\frac{\sum_{i\colon
G_{i}=1}Y_{i1}}{n_{11}}\right)-\left(\frac{\sum_{i\colon
G_{i}=0}Y_{i2}}{n_{02}}-\frac{\sum_{i\colon
G_{i}=0}Y_{i1}}{n_{01}}\right)\right\\}$
$\displaystyle\qquad-\left\\{\left(\frac{\sum_{i\colon
G_{i}=1}Y_{i1}}{n_{11}}-\frac{\sum_{i\colon
G_{i}=1}Y_{i0}}{n_{10}}\right)-\left(\frac{\sum_{i\colon
G_{i}=0}Y_{i1}}{n_{01}}-\frac{\sum_{i\colon
G_{i}=0}Y_{i0}}{n_{00}}\right)\right\\},$ (10)

where the first four terms are equal to the standard DID estimator (equation
(3)), and the last four terms are the DID estimator applied to pre-treatment
periods $t=0$ and $t=1$ (equation (4)).

This sequential DID estimator requires the parallel trends-in-trends
assumption — in the absence of the treatment, the change in the outcome trends
of the treatment group is equal to the change in the outcome trends of the
control group [32, e.g.,]. While the parallel trends assumption requires that
the outcome trends themselves are the same across the treatment and control
groups, the parallel trends-in-trends assumption only requires the change in
trends over time to be the same. Formally, the parallel trends-in-trends
assumption can be written as follows.

######  Assumption 3 (Parallel Trends-in-Trends).

$\displaystyle\underbrace{\big{\\{}\mathbb{E}[Y_{i2}(0)\mid
G_{i}=1]-\mathbb{E}[Y_{i1}(0)\mid G_{i}=1]\big{\\}}}_{\text{Trend of the
treatment group from $t=1$ to
$t=2$}}-\underbrace{\big{\\{}\mathbb{E}[Y_{i1}(0)\mid
G_{i}=1]-\mathbb{E}[Y_{i0}(0)\mid G_{i}=1]\big{\\}}}_{\text{Trend of the
treatment group from $t=0$ to $t=1$}}$ (11) $\displaystyle=$
$\displaystyle\underbrace{\big{\\{}\mathbb{E}[Y_{i2}(0)\mid
G_{i}=0]-\mathbb{E}[Y_{i1}(0)\mid G_{i}=0]\big{\\}}}_{\text{Trend of the
control group from $t=1$ to
$t=2$}}-\underbrace{\big{\\{}\mathbb{E}[Y_{i1}(0)\mid
G_{i}=0]-\mathbb{E}[Y_{i0}(0)\mid G_{i}=0]\big{\\}}}_{\text{Trend of the
control group from $t=0$ to $t=1$}}.\hskip 14.45377pt$

Here, the left-hand side represents how the outcome trends of the treatment
group change between (from $t=0$ to $t=1$) and (from $t=1$ to $t=2$). The
right-hand side quantifies the same change in the outcome trends for the
control group.

Figure 2: Comparing Extended Parallel Trends and Parallel Trends-in-Trends
Assumptions. Note: Below each panel, we report the trends of the control
potential outcomes for the treatment and control groups. The first and second
elements show the outcome trends (from $t=0$ to $t=1$) and (from $t=1$ to
$t=2$), respectively. The extended parallel trends assumption (left panel)
means that the outcome trends are the same across the treatment and control
groups for both (from $t=0$ to $t=1$) and (from $t=1$ to $t=2$). The parallel
trends-in-trends assumption (middle panel) only requires its change over time
is the same across the treatment and control groups;
$(-1)-(-2)=(-2.5)-(-3.5)=1$. Both assumptions are violated in the right panel.

We also emphasize an alternative way to interpret the parallel trends-in-
trends assumption. Unlike the parallel trends assumption that assumes the
time-invariant unmeasured confounding, the parallel trends-in-trends
assumption can account for linear time-varying unmeasured confounding —
unobserved confounding increases or decreases over time but with some constant
rate. We provide examples and formal justification of this interpretation in
Appendix C.3.3.

Figure 2 visually illustrates that the parallel trends-in-trends assumption
holds even when the trends of the treatment and control groups are not
parallel, as long as its change over time is the same. Under the parallel
trends-in-trends assumption, the sequential DID estimator is unbiased and
consistent for the ATT. Importantly, the extended parallel trends assumption
is stronger than the parallel trends-in-trends assumption, and thus, the
sequential DID estimator is unbiased and consistent for the ATT under the
extended parallel trends assumption as well.

We demonstrate that a common robustness check of including group- or unit-
specific time trends [3] is nonparametrically equivalent to the sequential DID
estimator (see Appendix C.3). Within the potential outcomes framework, we
clarified that these common techniques are justified under the parallel
trends-in-trends assumption.

##  3 Double Difference-in-Differences

We saw in the previous section that multiple pre-treatment periods provide the
three related benefits. We have clarified that each benefit requires different
assumptions and estimators, and as a result, in practice, researchers tend to
enjoy only a subset of the three benefits. In this section, we propose a new,
simple estimator, which we call the double difference-in-differences (double
DID), that blends all the three benefits of multiple pre-treatment periods in
a single framework. Here, we introduce the double DID with settings with two
pre-treatment periods.

We also provide three extensions. First, we propose the double DID regression
to include observed pre-treatment covariates (Section 3.3.1). Second, we
generalize the proposed method to any number of pre\- and post-treatment
periods in the DID design (Section 3.3.2). Finally, we extend it to the
staggered adoption design, where the timing of the treatment assignment can
vary across units (Section 4).

###  3.1 Double DID via Generalized Method of Moments

We propose the double DID estimator within a framework of the generalized
method of moments (GMM) [18]. In particular, we combine the standard DID
estimator and the sequential DID estimator via the GMM:

$\widehat{\tau}_{\texttt{d-DID}}=\operatornamewithlimits{argmin}_{\tau}\begin{pmatrix}\tau-\widehat{\tau}_{\texttt{DID}}\\\\[-5.0pt]
\tau-\widehat{\tau}_{\texttt{s-DID}}\end{pmatrix}^{\top}\mathbf{W}\begin{pmatrix}\tau-\widehat{\tau}_{\texttt{DID}}\\\\[-5.0pt]
\tau-\widehat{\tau}_{\texttt{s-DID}}\end{pmatrix}\vspace{0.05in}$ (12)

where $\mathbf{W}$ is a weight matrix of dimension $2\times 2$.

The important property of the proposed double DID estimator is that it
contains all of the popular estimators that we considered in the previous
sections as special cases. Table 1 illustrates that a particular choice of the
weight matrix $\mathbf{W}$ recovers the standard DID, the extended DID, and
the sequential DID estimators, respectively.

Using the GMM theory, we can estimate the optimal weight matrix
$\widehat{\mathbf{W}}$ such that asymptotic standard errors of the double DID
estimator are minimized, which we describe in detail in Section 3.1.2.
Therefore, users do not need to manually pick the weight matrix $\mathbf{W}$.

| Standard DID | Extended DID | Sequential DID
---|---|---|---
Weight Matrix | $\begin{pmatrix}1&0\\\ 0&0\end{pmatrix}$ | $\begin{pmatrix}3&0\\\ 0&-1\end{pmatrix}$ | $\begin{pmatrix}0&0\\\ 0&1\end{pmatrix}$
$\mathbf{W}$
Table 1: Double DID as Generalization of Popular DID Estimators.

We emphasize that the double DID estimator provides a unifying framework to
consider identification assumptions and to estimate treatment effects within
the framework of the GMM. The double DID estimator proceeds with the following
two steps.

####  3.1.1 Step 1: Assessing Underlying Assumptions

The first step is to assess the underlying assumptions. We use this first step
to adaptively choose the weight matrix $\mathbf{W}$ in the second step. In
this first step, we check the extended parallel trends assumption by applying
the DID estimator on pre-treatment periods (equation (4)) and testing whether
the estimate is statistically distinguishable from zero at a conventional
level. To take into account correlated errors, we cluster standard errors at
the level of treatment assignment.

Importantly, this step of the double DID can be viewed as the over-
identification test in the GMM framework [18], which tests whether all the
moment conditions are valid. In the context of the double DID estimator, we
assume that the sequential DID estimator is correctly specified and test the
null hypothesis that the standard DID estimator is correctly specified. Then,
the null hypothesis of the over-identification test becomes exactly the same
as testing whether an estimate of the DID estimator applied to pre-treatment
periods is equal to zero.

Equivalence Approach. We note that the standard hypothesis testing approach
has a risk of conflating evidence for parallel trends and statistical
inefficiency. For example, when sample size is small, even if pre-treatment
trends of the treatment and control groups differ, a test of the difference
might not be statistically significant due to large standard error, and
analysts might “pass” the pre-treatment-trends test. To mitigate such
concerns, we also incorporate an equivalence approach [19, e.g.,] in which we
evaluate the null hypothesis that trends of two groups are not parallel in
pre-treatment periods.444[29] propose a similar test for a different class of
estimators, what they refer to as “counterfactual estimators.” By using this
approach, researchers can “pass” the pre-treatment-trends test only when
estimated pre-treatment trends of the two groups are similar with high
accuracy, thereby avoiding the aforementioned common mistake. To facilitate
the interpretation of the equivalence confidence interval, we report the
standardized interval, which can be interpreted as the standard deviation from
the baseline control mean. We provide technical details in Appendix F and
provide an empirical example in Section 3.4.

####  3.1.2 Step 2: Estimation of the ATT

The second step is estimation of the ATT. When the extended parallel trends
assumption is plausible, we estimate the optimal weight matrix
$\widehat{\mathbf{W}}$ building on the theory of the efficient GMM [18].
Specifically, the optimal weight matrix that minimizes the variance of the
estimator is given by the inverse of the variance-covariance matrix of the two
DID estimators:

$\widehat{\mathbf{W}}=\begin{pmatrix}\widehat{{\rm
Var}}(\widehat{\tau}_{\texttt{DID}})&\widehat{\mbox{Cov}}(\widehat{\tau}_{\texttt{DID}},\widehat{\tau}_{\texttt{s-DID}})\\\
\widehat{\mbox{Cov}}(\widehat{\tau}_{\texttt{DID}},\widehat{\tau}_{\texttt{s-DID}})&\widehat{{\rm
Var}}(\widehat{\tau}_{\texttt{s-DID}})\end{pmatrix}^{-1}$ (13)

While the double DID approach can take any weight matrix, this optimal weight
matrix allows us to compute the weighted average of the standard DID and the
sequential DID estimator such that the resulting variance is the smallest. In
particular, when this optimal weight matrix is used, the double DID estimator
can be explicitly written as

$\widehat{\tau}_{\texttt{d-DID}}=w_{1}\widehat{\tau}_{\texttt{DID}}+w_{2}\widehat{\tau}_{\texttt{s-DID}}$
(14)

where $w_{1}+w_{2}=1$, and

$\displaystyle w_{1}$ $\displaystyle=\cfrac{\widehat{{\rm
Var}}(\widehat{\tau}_{\texttt{s-DID}})-\widehat{\mbox{Cov}}(\widehat{\tau}_{\texttt{DID}},\widehat{\tau}_{\texttt{s-DID}})}{\widehat{{\rm
Var}}(\widehat{\tau}_{\texttt{DID}})+\widehat{{\rm
Var}}(\widehat{\tau}_{\texttt{s-DID}})-2\widehat{\mbox{Cov}}(\widehat{\tau}_{\texttt{DID}},\widehat{\tau}_{\texttt{s-DID}})},$
$\displaystyle w_{2}$ $\displaystyle=\cfrac{\widehat{{\rm
Var}}(\widehat{\tau}_{\texttt{DID}})-\widehat{\mbox{Cov}}(\widehat{\tau}_{\texttt{DID}},\widehat{\tau}_{\texttt{s-DID}})}{\widehat{{\rm
Var}}(\widehat{\tau}_{\texttt{DID}})+\widehat{{\rm
Var}}(\widehat{\tau}_{\texttt{s-DID}})-2\widehat{\mbox{Cov}}(\widehat{\tau}_{\texttt{DID}},\widehat{\tau}_{\texttt{s-DID}})}.$

By pooling information from both the standard DID and sequential DID, the
asymptotic variance of the double DID is smaller than or equal to variance of
either the standard and sequential DIDs. This is analogous to Bayesian
hierarchical models where pooling information from multiple groups makes
estimation more accurate than separate estimation based on each group.

In addition, because the extended DID is a special case of the double DID (as
described in Table 1), the asymptotic variance of the double DID is also
smaller than or equal to variance of the extended DID. Therefore, ${\rm
Var}(\widehat{\tau}_{\texttt{d-DID}})\leq\mbox{min}({\rm
Var}(\widehat{\tau}_{\texttt{DID}}),{\rm
Var}(\widehat{\tau}_{\texttt{s-DID}}),{\rm
Var}(\widehat{\tau}_{\texttt{e-DID}})).$ We provide the proof in Appendix D.

Following [9], we estimate the variance-covariance matrix of
$\widehat{\tau}_{\texttt{DID}}$ and $\widehat{\tau}_{\texttt{s-DID}}$ via
block-bootstrap where the block is taken at the level of treatment assignment.
Specifically, we obtain a pair of two estimators
$\\{\widehat{\tau}^{(b)}_{\texttt{DID}},\widehat{\tau}^{(b)}_{\texttt{s-DID}}\\}$
for $b=1,\ldots,B$ with $B$ number of bootstrap iterations, and compute the
empirical variance-covariance matrix. Given an estimate of the weight matrix
(equation (13)), we obtain the double DID estimate as a weighted average
(equation (14)). We can obtain the variance estimate of
$\widehat{\tau}_{\texttt{d-DID}}$ by following the standard efficient GMM
variance formula:

$\widehat{\text{Var}}(\widehat{\tau}_{\texttt{d-DID}})=(\bm{1}^{\top}\widehat{\mathbf{W}}\bm{1})^{-1},$

where $\bm{1}$ is a two-dimensional vector of ones.

##### Remark.

Under the extended parallel trends assumption, both the standard DID and the
sequential DID estimator are consistent for the ATT, and thus, any weighted
average is a consistent estimator. But the optimal weight matrix (equation
(13)) chooses the most efficient estimator among all consistent estimators. As
we clarify more below, we do not use the weighted average of the standard DID
and the sequential DID when the extended parallel trends assumption is
violated. ∎

When only the parallel trends-in-trends assumption is plausible, the double
DID contains one moment condition $\tau-\widehat{\tau}_{\texttt{s-DID}}=0$,
and thus, it reduces to the sequential DID estimator. This is equivalent to
choosing the weight matrix $\mathbf{W}$ with $W_{11}=W_{12}=W_{21}=0$ and
$W_{22}=1$ (the third column in Table 1).

When both assumptions are implausible, there is no credible estimator for the
ATT without making further stringent assumptions. However, when there are more
than two pre-treatment periods, researchers can also use the proposed
generalized $K$-DID (discussed in Section 3.3.2) to further relax the parallel
trends-in-trends assumption.

###  3.2 Double DID Enjoys Three Benefits

The proposed double DID estimator naturally enjoys the three benefits of
multiple pre-treatment periods within a unified framework.

##### 1\. Assessing Underlying Assumptions

The double DID incorporates the assessment of underlying assumptions in its
first step as the over-identification test. When the trends in pre-treatment
periods are not parallel, researchers have to pay the most careful attention
to research design and use domain knowledge to assess the parallel trends-in-
trends assumption.

##### 2\. Improving Estimation Accuracy

When the extended parallel trends assumption holds, researchers can combine
two DIDs with equal weights (i.e., the extended DID estimator, which is
numerically equivalent to the two-way fixed effects regression) to increase
estimation accuracy (Section 2.3). In this setting, the double DID further
improves estimation accuracy because it selects the optimal weights as the GMM
estimator. In Section G, we use simulations to show that the double DID
achieves smaller standard errors than the extended DID estimator.

##### 3\. Allowing For A More Flexible Parallel Trends Assumption

Under the parallel trends-in-trends assumption, the double DID estimator
converges to the sequential DID estimator. However, when the extended parallel
trends assumption holds, the double DID uses optimal weights and is not equal
to the sequential DID. Thus, the double DID estimator avoids a dilemma of the
sequential DID — it is consistent under a weaker assumption of the parallel
trends-in-trends but is less efficient when the extended parallel trends
assumption holds. By naturally changing the weight matrix in the GMM
framework, the double DID achieves high estimation accuracy under the extended
parallel trends assumption and, at the same time, allows for more flexible
time-varying unmeasured confounding under the parallel trends-in-trends
assumption.

###  3.3 Extensions

####  3.3.1 Double DID Regression

Like other DID estimators, the double DID estimator has a nice connection to a
regression approach. We propose the double DID regression with which
researchers can include other pre-treatment covariates $\mathbf{X}_{it}$ to
make the DID design more robust and efficient. We provide technical details in
Appendix E.1.

####  3.3.2 Generalized $K$-Difference-in-Differences

We generalize the proposed method to _any_ number of pre- and post-treatment
periods in Appendix E.2, which we call $K$-difference-in-differences
($K$-DID). This generalization has two central benefits. First, it enables
researchers to use longer pre-treatment periods to allow for even more
flexible forms of unmeasured time-varying confounding beyond the linear time-
varying unmeasured confounding under the parallel trends-in-trends assumption
(Assumption 3). $K$-DID allows for time-varying unmeasured confounding that
follows a $(K-1)$th order polynomial function when researchers have $K$ pre-
treatment periods. We can view the double DID as a special case of $K$-DID
because in the double DID we have $K=2$ pre-treatment periods, and it can
allow for unmeasured confounding that follows ($2-1=1$)st order polynomial
function (i.e., a linear function).

Second, we also allow for any number of post-treatment periods so that
researchers can estimate not only short-term causal effects but also longer-
term causal effects. This generalization can be crucial in many applications
because treatment effects might not have an immediate impact on the outcome.

###  3.4 Empirical Application

[30] utilize the basic DID design to study how the abolition of elected
councils affects local public services in Vietnam. To estimate the causal
effects of the institutional change, the original authors rely on data from
2008 and 2010, which are before and after the abolition of elected councils in
2009\. Then, they supplement the main analysis by assessing trends in pre-
treatment periods from 2006 to 2008. In this section, we apply the proposed
method and illustrate how to improve this basic DID design.

Although [30] employ the exact same DID design to all of the thirty outcomes
they consider, each outcome might require different assumptions, as noted in
the original paper. Here, we focus on reanalyzing three outcomes that have
different patterns of pre-treatment periods. By doing so, we clarify how
researchers can use the double DID method to transparently assess underlying
assumptions and employ appropriate DID estimators under different settings. We
provide an analysis of all thirty outcomes in Appendix H.1.

####  3.4.1 Visualizing and Assessing Underlying Assumptions

The first step of the DID design is to visualize trends of treatment and
control groups. Figure 3 shows trends of three different outcomes: “Education
and Cultural Program,” “Tap Water,” and “Agricultural Center.”555See Appendix
H.1 for definitions. Although the original analysis uses the same DID design
for all of them, they have distinct trends in the pre-treatment periods. The
first outcome of “Education and Cultural Program” has parallel trends in pre-
treatment periods. For the other two outcomes, trends do not look parallel in
either of the cases. While the trends for the second outcome (“Tap Water”)
have similar directions, trends for the third outcome (“Agricultural Center”)
have opposite signs. This visualization of trends serves as a transparent
first step to assess the underlying assumptions necessary for the DID
estimation.

Figure 3: Visualizing Trends of Treatment and Control Groups. Note: We report
trends for the treatment group (black solid line with solid circles) and the
control group (gray dashed line with hollow circles). Two pre-treatment
periods are 2006 and 2008. One post-treatment period, 2010, is indicated by
the gray shaded area.

The next step is to formally assess underlying assumptions. As in the original
study, it is common to incorporate additional covariates to make the parallel
trends assumption more plausible. Based on detailed domain knowledge, [30]
include four control variables: area size of each commune, population size,
whether national-level city or not, and regional fixed effects. Thus, we
assess the conditional extended parallel trends assumption by fitting the DID
regression to pre-treatment periods from 2006 to 2008, where $\mathbf{X}_{it}$
includes the four control variables. If the conditional extended parallel
trends assumption holds, estimates of the DID regression on pre-treatment
trends should be close to zero.

While a traditional approach is to assess whether estimates are statistically
distinguishable from zero with the conventional 5% or 10% level, we also
report results based on an equivalence approach that we recommend in Section
3. Specifically, we compute the 95% standardized equivalence confidence
interval, which quantifies the smallest equivalence range supported by the
observed data [19]. In the context of this application, the equivalence
confidence interval is standardized based on the mean and standard deviation
of the control group in 2006. For example, if the 95% standardized equivalence
confidence interval is $[-\nu,\nu],$ this means that the equivalence test
rejects the hypothesis that the DID estimate (standardized with respect to the
baseline control outcome) on pre-treatment periods is larger than $\nu$ or
smaller than $-\nu$ at the 5% level. Thus, the conditional extended parallel
trends assumption is more plausible when the equivalence confidence interval
is shorter.

The results are summarized in Table 2. Standard errors are computed via block-
bootstrap at the district level, where we take 2000 bootstrap iterations. For
the first outcome, as the graphical presentation in Figure 3 suggests, a
statistical test suggests that the extended parallel trends assumption is
plausible.

For the second outcome, the test of the parallel trends reveals that the
parallel trends assumption is less plausible for this outcome than for the
first outcome. Finally, for the third outcome, both traditional and
equivalence approaches provide little evidence for parallel trends, as
graphically clear in Figure 3. Although we only have two pre-treatment periods
as in the original analysis, if more than two pre-treatment periods are
available, researchers can assess the extended parallel trends-in-trends
assumption in a similar way by applying the sequential DID estimator to pre-
treatment periods. Upon assessing the underlying parallel trends assumptions,
we now proceed to estimation of the ATT via the double DID.

| Estimate | Std. Error | p-value | 95% Std. Equivalence CI
---|---|---|---|---
Education and Cultural Program | $-0.007$ | 0.096 | 0.940 | $[-0.166,0.166]$
Tap Water | 0.166 | 0.083 | 0.045 | $[-0.302,0.302]$
Agricultural Center | 0.198 | 0.082 | 0.015 | $[-0.332,0.332]$
Table 2: Assessing Underlying Assumptions Using the Pre-treatment Outcomes.
Note: We evaluate the conditional extended parallel trends assumption for
three different outcomes. The table reports DID estimates on pre-treatment
trends, standard errors, p-values, and the 95% standardized equivalence
confidence intervals.

####  3.4.2 Estimating Causal Effects

Within the double DID framework, we select appropriate DID estimators after
the empirical assessment of underlying assumptions. For the first outcome,
diagnostics in the previous section suggest that the extended parallel trends
assumption is plausible. In such settings, the double DID is expected to
produce similar point estimates with smaller standard errors compared to the
conventional DID estimator. The first plot of Figure 4 clearly shows this
pattern. In the figure, we report point estimates as well as 90% confidence
intervals following the original paper (see Figure 3 in [30]). Using the
standard DID estimator, the original estimate of the ATT on “Education and
Cultural Program” was $0.084$ (90% CI = [$-0.006,0.174$]). Using the double
DID estimator, an estimate is instead $0.082$ (90% CI = [$0.001,0.163$]). By
using the double DID estimator, we shrink standard errors by about 10%.
Although we only have two pre-treatment periods here, when there are more pre-
treatment periods, efficiency gain of the double DID can be even larger.

Figure 4: Estimating Causal Effects of Abolishing Elected Councils. Note: We
compare estimates from the standard DID and the proposed double DID.

For the second outcome, we did not have enough evidence to support the
extended parallel trends assumption. Thus, instead of using the standard DID
as in the original analysis, we rely on the parallel trends-in-trends
assumption. In this case, the double DID estimates the ATT by allowing for
linear time-varying unmeasured confounding in contrast to the standard DID
that assumes constant unmeasured confounders. The second plot of Figure 4
shows the important difference between the two methods. While the standard DID
estimates is $-0.078$ (90% CI = [$-0.169,0.012$]), the double DID estimate is
$-0.119$ (90% CI = [$-0.225,-0.012$]). Given that the extended parallel trends
assumption is not plausible, this result suggests that the standard DID
suffers from substantial bias (the bias of $0.04$ corresponds to more than 50%
of the original point estimate). By incorporating non-parallel pre-treatment
trends, the double DID shows that the original DID estimate was underestimated
by a large amount.

Finally, for the third outcome, the previous diagnostics suggest that the
extended parallel trends assumption is implausible. It is possible to use the
double DID under the parallel trends-in-trends assumption. However, trends of
treatment and control groups have opposite signs, implying the double DID
estimates are highly sensitive to the parallel trends-in-trends assumption.
Given that the parallel trends-in-trends assumption is also difficult to
justify here, there is no credible estimator of the ATT without making
additional stringent assumptions. While we focused on the three outcomes here,
the double DID improves upon the standard DID in a similar way for the other
outcomes as well (see Appendix H.1).

##  4 Staggered Adoption Design

In this section, we extend the proposed double DID estimator to the staggered
adoption design where the timing of the treatment assignment can vary across
units [36, 7, 5].

###  4.1 The Setup and Causal Quantities of Interest

In the staggered adoption (SA) design, different units can receive the
treatment in different time periods. Once they receive the treatment, they
remain exposed to the treatment afterward. Therefore, $D_{it}=1$ if $D_{im}=1$
where $m<t.$ We can thus summarize information about the treatment assignment
by the timing of the treatment $A_{i}$ where $A_{i}\equiv\mbox{min}\
\\{t:D_{it}=1\\}.$ When unit $i$ never receives the treatment until the end of
time $T$, we let $A_{i}=\infty.$ For example, in many applications where
researchers are interested in the causal effect of state- or local-level
policies, units adopt policies in different time points and remain exposed to
such policies once they introduce the policies. In Appendix H.2, we provide
its example based on [34]. See Figure 5 for visualization of the SA design.

Following the recent literature on the SA design, we make two standard
assumptions in the SA design: no anticipation assumption and invariance to
history assumption [23, 5]. This implies that, for unit $i$ in period $t$, the
potential outcome $Y_{it}(1)$ represents the outcome of unit $i$ that would
realize in period $t$ if unit $i$ receives the treatment at or before period
$t$. Similarly, $Y_{it}(0)$ represents the outcome of unit $i$ that would
realize in period $t$ if unit $i$ does not receive the treatment by period
$t$. Finally, we generalize group indicator $G$ as follows.

$G_{it}\ =\ \begin{cases}\ \ 1&\mbox{if }A_{i}=t\\\ \ \ 0&\mbox{if }A_{i}>t\\\
\ \ -1&\mbox{if }A_{i}<t\end{cases}$ (15)

where $G_{it}=1$ represents units who receive the treatment at time $t$, and
$G_{it}=0$ ($G_{it}=-1$) indicates units who receive the treatment after
(before) time $t$.

Under the SA design, the staggered adoption ATT (SA-ATT) at time $t$ is
defined as follows.

$\tau^{\textsf{SA}}(t)=\mathbb{E}[Y_{it}(1)-Y_{it}(0)\mid G_{it}=1],$

which represents the causal effect of the treatment in period $t$ on units
with $G_{it}=1$, who receive the treatment at time $t.$ This is a
straightforward extension of the standard ATT (equation (1)) in the basic DID
setting. Researchers might also be interested in the time-average staggered
adoption ATT (time-average SA-ATT).

$\overline{\tau}^{\textsf{SA}}=\sum_{t\in\mathcal{T}}\pi_{t}\tau^{\textsf{SA}}(t),$

where $\mathcal{T}$ represents a set of the time periods for which researchers
want to estimate the ATT. For example, if a researcher is interested in
estimating the ATT for the entire sample periods, one can take
$\mathcal{T}=\\{1,\ldots,T\\}$. The SA-ATT in period $t$,
$\tau^{\textsf{SA}}(t),$ is weighted by the proportion of units who receive
the treatment at time $t$:
$\pi_{t}=\sum_{i=1}^{n}\mathbf{1}\\{A_{i}=t\\}/\sum_{i=1}^{n}\mathbf{1}\\{A_{i}\in\mathcal{T}\\}$.

Figure 5: Example of the Staggered Adoption Design. Note: We use gray cells of
“1” to denote the treated observation and use white cells of “0” to denote the
control observation.

###  4.2 Double DID for Staggered Adoption Design

Under what assumptions can we identify the SA-ATT and the time-average SA-ATT?
Here, we first extend the standard DID estimator under the parallel trends
assumption and the sequential DID estimator under the parallel trends-in-
trends assumption to the SA design. Formally, we define the standard DID
estimator for the SA-ATT at time $t$ as

$\widehat{\tau}^{\textsf{SA}}_{\texttt{DID}}(t)=\left(\frac{\sum_{i\colon
G_{it}=1}Y_{it}}{n_{1t}}-\frac{\sum_{i\colon
G_{it}=1}Y_{i,t-1}}{n_{1,t-1}}\right)-\left(\frac{\sum_{i\colon
G_{it}=0}Y_{it}}{n_{0t}}-\frac{\sum_{i\colon
G_{it}=0}Y_{i,t-1}}{n_{0,t-1}}\right),\vspace{10pt}$

which is consistent for the SA-ATT under the following parallel trends
assumption in period $t$ under the SA design:

$\mathbb{E}[Y_{it}(0)\mid G_{it}=1]-\mathbb{E}[Y_{i,t-1}(0)\mid G_{it}=1]\ =\
\mathbb{E}[Y_{it}(0)\mid G_{it}=0]-\mathbb{E}[Y_{i,t-1}(0)\mid G_{it}=0].$

Similarly, we can define the sequential DID estimator for the SA-ATT at time
$t$ as

$\displaystyle\widehat{\tau}^{\textsf{SA}}_{\texttt{s-DID}}(t)$
$\displaystyle=\left\\{\left(\frac{\sum_{i\colon
G_{it}=1}Y_{it}}{n_{1t}}-\frac{\sum_{i\colon
G_{it}=1}Y_{i,t-1}}{n_{1,t-1}}\right)-\left(\frac{\sum_{i\colon
G_{it}=0}Y_{it}}{n_{0t}}-\frac{\sum_{i\colon
G_{it}=0}Y_{i,t-1}}{n_{0,t-1}}\right)\right\\}$
$\displaystyle\qquad-\left\\{\left(\frac{\sum_{i\colon
G_{it}=1}Y_{i,t-1}}{n_{1,t-1}}-\frac{\sum_{i\colon
G_{it}=1}Y_{i,t-2}}{n_{1,t-2}}\right)-\left(\frac{\sum_{i\colon
G_{it}=0}Y_{i,t-1}}{n_{0,t-1}}-\frac{\sum_{i\colon
G_{it}=0}Y_{i,t-2}}{n_{0,t-2}}\right)\right\\},$

which is consistent for the SA-ATT under the following parallel trends-in-
trends assumption in period $t$ under the SA design:

$\displaystyle\\{\mathbb{E}[Y_{it}(0)\mid G_{it}=1]-\mathbb{E}[Y_{it}(0)\mid
G_{it}=0]\\}-\\{\mathbb{E}[Y_{i,t-1}(0)\mid
G_{it}=1]-\mathbb{E}[Y_{i,t-1}(0)\mid G_{it}=0]\\}$
$\displaystyle=\\{\mathbb{E}[Y_{i,t-1}(0)\mid
G_{it}=1]-\mathbb{E}[Y_{i,t-1}(0)\mid
G_{it}=0]\\}-\\{\mathbb{E}[Y_{i,t-2}(0)\mid
G_{it}=1]-\mathbb{E}[Y_{i,t-2}(0)\mid G_{it}=0]\\}.$

Finally, combining the standard and sequential DID estimators, we can extend
the double DID to the SA design as follows.

$\widehat{\tau}^{\textsf{SA}}_{\texttt{d-DID}}(t)=\operatornamewithlimits{argmin}_{\tau^{\textsf{SA}}(t)}\begin{pmatrix}\tau^{\textsf{SA}}(t)-\widehat{\tau}^{\textsf{SA}}_{\texttt{DID}}(t)\\\\[-5.0pt]
\tau^{\textsf{SA}}(t)-\widehat{\tau}^{\textsf{SA}}_{\texttt{s-DID}}(t)\end{pmatrix}^{\top}\mathbf{W}(t)\begin{pmatrix}\tau^{\textsf{SA}}(t)-\widehat{\tau}^{\textsf{SA}}_{\texttt{DID}}(t)\\\\[-5.0pt]
\tau^{\textsf{SA}}(t)-\widehat{\tau}^{\textsf{SA}}_{\texttt{s-DID}}(t)\end{pmatrix}\vspace{0.05in}$

where $\mathbf{W}(t)$ is a weight matrix. Under the SA design, similar to the
basic design, the standard DID and sequential DID estimators are special cases
of our proposed double DID estimator with specific choices of the weight
matrix. As in Section 3.1, we can estimate the optimal weight matrix
$\widehat{\mathbf{W}}(t)$ (details below), and thus, users do not need to
choose it manually.

Like the basic double DID estimator in Section 3.1, the double DID for the SA
design also consists of two steps. The first step is to assess the underlying
assumptions using the standard DID for the SA design with two points
$\\{t-1,t-2\\}$ for units that are not yet treated at time $t-1$, that is,
$\\{i:G_{it}\geq 0\\}$. This is a generalization of the pre-treatment-trends
test in the basic DID setup (Section 2.2). The second step is to estimate the
SA-ATT at time $t$. When only the parallel trends-in-trends assumption is
plausible, we choose weight matrix $\mathbf{W}(t)$ where
$\mathbf{W}(t)_{11}=\mathbf{W}(t)_{12}=\mathbf{W}(t)_{21}=0$ and
$\mathbf{W}(t)_{22}=1$, which converges to the sequential DID under the SA
design. When the extended parallel trends assumption is plausible, we use the
optimal weight matrix defined as $\widehat{\mathbf{W}}(t)=\widehat{{\rm
Var}}(\widehat{\tau}^{\textsf{SA}}_{(1:2)}(t))^{-1}$ where ${\rm Var}(\cdot)$
is the variance-covariance matrix and
$\widehat{\tau}^{\textsf{SA}}_{(1:2)}(t)=(\widehat{\tau}^{\textsf{SA}}_{\texttt{DID}}(t),\widehat{\tau}^{\textsf{SA}}_{\texttt{s-DID}}(t))^{\top}.$
This optimal weight matrix provides us with the most efficient estimator
(i.e., the smallest standard error). We provide further details on the
implementation in Appendix E.3.

To estimate the time-average SA-DID, we extend the double DID as follows.

$\widehat{\overline{\tau}}^{\textsf{SA}}_{\texttt{d-DID}}=\operatornamewithlimits{argmin}_{\overline{\tau}^{\textsf{SA}}}\begin{pmatrix}\overline{\tau}^{\textsf{SA}}-\widehat{\overline{\tau}}^{\textsf{SA}}_{\texttt{DID}}\\\\[-5.0pt]
\overline{\tau}^{\textsf{SA}}-\widehat{\overline{\tau}}^{\textsf{SA}}_{\texttt{s-DID}}\end{pmatrix}^{\top}\overline{\mathbf{W}}\begin{pmatrix}\overline{\tau}^{\textsf{SA}}-\widehat{\overline{\tau}}^{\textsf{SA}}_{\texttt{DID}}\\\\[-5.0pt]
\overline{\tau}^{\textsf{SA}}-\widehat{\overline{\tau}}^{\textsf{SA}}_{\texttt{s-DID}}\end{pmatrix}\vspace{0.05in}$

where $\widehat{\overline{\tau}}^{\textsf{SA}}_{\texttt{DID}}$ and
$\widehat{\overline{\tau}}^{\textsf{SA}}_{\texttt{s-DID}}$ are time-averages
of the DID and sequential DID estimators,

$\widehat{\overline{\tau}}^{\textsf{SA}}_{\texttt{DID}}=\sum_{t\in\mathcal{T}}\pi_{t}\widehat{\tau}^{\textsf{SA}}_{\texttt{DID}}(t),\hskip
14.45377pt\mbox{and}\hskip
14.45377pt\widehat{\overline{\tau}}^{\textsf{SA}}_{\texttt{s-DID}}=\sum_{t\in\mathcal{T}}\pi_{t}\widehat{\tau}^{\textsf{SA}}_{\texttt{s-DID}}(t).$

The optimal weight matrix $\widehat{\overline{\mathbf{W}}}$ is equal to
$\widehat{{\rm Var}}(\widehat{\overline{\tau}}^{\textsf{SA}}_{(1:2)})^{-1}$
where
$\widehat{\overline{\tau}}^{\textsf{SA}}_{(1:2)}=(\widehat{\overline{\tau}}^{\textsf{SA}}_{\texttt{DID}},\widehat{\overline{\tau}}^{\textsf{SA}}_{\texttt{s-DID}})^{\top}.$

##  5 Concluding Remarks

While the most basic form of the DID only requires two time periods — one
before and the other after treatment assignment, researchers can often collect
data from several additional pre-treatment periods in a wide range of
applications. In this article, we show that such multiple pre-treatment
periods can help improve the basic DID design and the staggered adoption
design in three ways: (1) assessing underlying assumptions about parallel
trends, (2) improving estimation accuracy, and (3) enabling more flexible DID
estimators. We use the potential outcomes framework to clarify assumptions
required to enjoy each benefit.

We then propose a simple method, the double DID, to combine all three benefits
within the GMM framework. Importantly, the double DID contains the popular
two-way fixed effects regression and nonparametric DID estimators as special
cases, and it uses the GMM to further improve with respect to identification
and estimation accuracy. Finally, we generalize the double DID estimator to
the staggered adoption design where the timing of the treatment assignment can
vary across units.

References.1References.1ReferencesReferencesReferences.1

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##  Appendix A Literature Review

###  A.1 Papers in APSR and AJPS

We conduct a review of the literature to assess current practices of the
difference-in-differences (DID) design. Specifically, we search articles
published in American Political Science Review and American Journal of
Political Science from 2015 to 2019. Some of the papers we reviewed were
accepted in 2019 and were officially published in 2020. Using Google Scholar,
we find articles that contains any of the following keywords: “two-way fixed
effect”, “two-way fixed effects”, “difference in difference” or “difference in
differences.” We then manually select articles from the list that uses the
basic DID design and the staggered adoption design (see the main text for
details about the first two design). This procedure left us with a total of 25
articles, 11 from APSR and 14 from AJPS. Table A1 and A2 show the articles in
the list published in APSR and AJPS, respectively.

To determine the number of pre-treatment periods, we manually assess the
listed articles. Among the 25 articles, 20 articles use the basic DID design,
and 5 articles use the staggered adoption design. When a paper uses the basic
DID design, we can determine the length of the pre-treatment periods from the
data description and the time of the treatment assignment. On the other hand,
the pre-treatment periods for the staggered adoption and the general design
are set to the total number of time-periods available in the data, as the
length of pre-treatment periods varies across units.

We found that most DID applications have less than 10 pre-treatment periods.
The median number of pre-treatment periods is $3.5$ and, the mean number of
pre-treatment periods is about $6$ after removing one unique study that has
more than $100$ pre-treatment periods.

###  A.2 Examples of Two Common Approaches

As we wrote in Section 1, there are several different popular ways to analyze
the DID design with multiple pre-treatment periods. One common approach is to
apply the two-way fixed effects regression to the entire time periods, and
supplement it with alternative model specifications by including time-trends
or leads of the treatment variable to assess possible violations of the
parallel trends assumption. Examples include [13, 38, 14, 17, 27]. Another is
to stick with the two-time-period DID and limit the use of additional pre-
treatment periods only to the assessment of pre-treatment trends. Examples
include [26, 6, 10, 25, 15]. Note that we list exemplary papers here and thus,
we also include papers from journals other than APSR and AJPS.

Table A1: DID papers on APSR. Authors | Year |  Title
---|---|---
O’brien, D. Z., & Rickne J. | 2016 |  Gender Quotas And Women’s Political Leadership
Garfias, F. | 2018 |  Elite Competition and State Capacity Development: Theory and Evidence From Post-Revolutionary Mexico.
Martin, G. J., & Mccrain, J. | 2019 |  Local News And National Politics
Blom-Hansen, J., Houlberg, K., Serritzlew, S., & Treisman, D. | 2016 |  Jurisdiction Size and Local Government Policy Expenditure: Assessing The Effect of Municipal Amalgamation
Clinton, J. D., & Sances, M. W. | 2018 |  The Politics of Policy: The Initial Mass Political Effects of Medicaid Expansion in The States
Malesky, E. J. , Nguyen, C. V., & Tran, A. | 2014 |  The Impact of Recentralization on Public Services: A Difference-in-Differences Analysis of the Abolition of Elected Councils in Vietnam.
Larsen, M. V., Hjorth, F., Dinesen, P. T., & Sønderskov, K. M. | 2019 |  When Do Citizens Respond Politically to The Local Economy? Evidence From Registry Data on Local Housing Markets
Becher, M., & González, I. M. | 2019 |  Electoral Reform and Trade-Offs in Representation
Selb, P., & Munzert, S. | 2018 |  Examining A Most Likely Case for Strong Campaign Effects
Enos, R. D., Kaufman, A. R., & Sands, M. L. | 2019 |  Can Violent Protest Change Local Policy Support?
Vasiliki Fouka | 2019 |  How Do Immigrants Respond to Discrimination?
Table A2: DID papers on AJPS. Authors | Year |  Title
---|---|---
Bechtel, M. M., Hangartner, D., & Schmid, L. | 2016 |  Does compulsory voting increase support for leftist policy?
Bisgaard, M., & Slothuus, R. | 2018 |  Partisan elites as culprits? How party cues shape partisan perceptual gaps.
Bischof, D., & Wagner, M. | 2019 |  Do voters polarize when radical parties enter parliament?
Dewan, T., Meriläinen, J., & Tukiainen, J. | 2020 |  Victorian voting: The origins of party orientation and class alignment.
Earle, J. S., & Gehlbach, S. | 2015 |  The Productivity Consequences of Political Turnover: Firm-Level Evidence from Ukraine’s Orange Revolution.
Enos, R. D. | 2016 |  What the demolition of public housing teaches us about the impact of racial threat on political behavior.
Gingerich, D. W. | 2019 |  Ballot Reform as Suffrage Restriction: Evidence from Brazil’s Second Republic.
Hainmueller, J, & Hangartner, D. | 2019 |  Does direct democracy hurt immigrant minorities? Evidence from naturalization decisions in Switzerland.
Holbein, J. B., & Hillygus, D. S. | 2016 |  Making young voters: the impact of preregistration on youth turnout.
Jäger, K. | 2020 |  When Do Campaign Effects Persist for Years? Evidence from a Natural Experiment.
Lindgren, K. O., Oskarsson, S., & Dawes, C. T. | 2017 |  Can Political Inequalities Be Educated Away? Evidence from a Large-Scale Reform.
Lopes da Fonseca, M. | 2017 |  Identifying the source of incumbency advantage through a constitutional reform.
Paglayan, AS. | 2019 |  Public-Sector Unions and the Size of Government
Pardos-Prado, S., & Xena, C. | 2019 |  Skill specificity and attitudes toward immigration.

##  Appendix B Comparison with Three Existing Methods

This section clarifies relationships between our proposed double DID and three
existing methods: the two-way fixed effects estimator, the sequential DID
estimator, and synthetic control methods.

###  B.1 Relationship with Two-Way Fixed Effects Estimator

While we contrast the double DID with the two-way fixed effects estimator
throughout the paper, we summarize our discussion here. First, in the basic
DID design, the two-way fixed effects estimator is a special case of the
double DID with a specific choice of the weight matrix $\mathbf{W}$ (see Table
1). Therefore, whenever the two-way fixed effects estimator is consistent for
the ATT, the double DID is a more efficient, consistent estimator of the ATT.
This is because the double DID can choose the optimal weight matrix via the
GMM, while the two-way fixed effects uses the pre-determined equal weights
over time. Second, in the SA design, a large number of recent papers show that
the widely-used two-way fixed effects estimator are in general inconsistent
for the ATT due to treatment effect heterogeneity and implicit parametric
assumptions [36, 5, 22, 37]. In contrast, the proposed double DID in the SA
design generalizes nonparametric DID estimators to allow for treatment effect
heterogeneity, and thus, it does not suffer from the same problem.

###  B.2 Relationship with Sequential DID Estimator

Our double DID estimator contains the sequential DID estimator [28, 32, e.g.,]
as a special case. Our proposed double DID improves over the sequential DID
estimator in two ways. First, when the parallel trends assumption holds, the
double DID optimally combine the standard DID and the sequential DID to
improve efficiency, and it is not equal to the sequential DID. Therefore, it
avoids a dilemma of the sequential DID — it is consistent under the parallel
trends-in-trends assumption (weaker than the parallel trends assumption), but
is less efficient when the parallel trends assumption holds. Second, while the
sequential DID estimator has only been available for the basic DID design
where treatment assignment happens only once, we generalize it to the
staggered adoption design and further incorporate it into our staggered-
adoption double DID estimator (Section 4).

###  B.3 Relationship with Synthetic Control Methods

Another relevant popular class of methods is the synthetic control methods.
While the method was originally designed to estimate the causal effect on a
single treated unit, recent extensions allow for multiple treated units and
the staggered adoption design [40, 8, 20, 4, e.g.,]. Despite a wide variety of
innovative extensions, they all share the same core feature: they require long
pre-treatment periods to accurately estimate a pre-treatment trajectory of the
treated units. For example, [40] recommends collecting more than ten pre-
treatment periods. In contrast, the proposed double DID can be applied as long
as there are more than one pre-treatment periods, and is better suited when
there are a small to moderate number of pre-treatment periods.

When there are a large number of pre-treatment periods (i.e., long enough to
apply the synthetic control methods), we recommend to apply both the synthetic
control methods and proposed double DID, and evaluate robustness across those
approaches. This is important because they rely on different identification
assumptions. In fact, we show in Section H.2, the double DID can recover
credible estimates similar to more flexible variants of synthetic control
methods even when there are a large number of pre-treatment periods. This
robustness provides researchers with additional credibility for their causal
estimates and underlying assumptions.

##  Appendix C Nonparametric Equivalence to Regression Estimators

In this section, we provide results on the nonparametric connection between
regression estimators and the three DID estimators we discussed in the paper.
This section provides methodological foundations for our main methodological
contributions, which we prove in Sections E.2 and E.3.

###  C.1 Standard DID

In practice, we can compute the DID estimator via a linear regression. We
regress the outcome $Y_{it}$ on an intercept, treatment group indicator
$G_{i}$, time indicator $I_{t}$ (equal to $1$ if post-treatment and $0$
otherwise) and the interaction between the treatment group indicator and the
time indicator $G_{i}\times I_{t}$.

$Y_{it}\sim\alpha+\theta G_{i}+\gamma I_{t}+\beta(G_{i}\times I_{t}),$ (A.2)

where $(\alpha,\theta,\gamma,\beta)$ are corresponding coefficients. In this
case, a coefficient of the interaction term $\beta$ is numerically equal to
the DID estimator, $\widehat{\tau}_{\texttt{DID}}$. Importantly, the linear
regression is used here only to compute the nonparametric DID estimator
(equation (3)), and thus it does not require any parametric modeling
assumption such as constant treatment effects. Furthermore, when we analyze
panel data in which the same units are observed repeatedly over time, we
obtain exactly the same estimate via a linear regression with unit and time
fixed effects. This numerical equivalence in the two-time-period case is often
the justification of the two-way fixed effects regression as the DID design
[3]. The above equivalence is formally shown below for completeness.

####  C.1.1 Repeated Cross-Sectional Data

For the later use in this Appendix, we report the well-known result that the
standard DID estimator $\widehat{\tau}_{\texttt{DID}}$ (equation (3)) is
equivalent to coefficient $\widehat{\beta}$ in the regression estimator
(equation (A.2)) [1].

We define $O_{it}$ to be an indicator variable taking the value $1$ when
individual $i$ is observed in time period $t$. Using this notation, we prove
the following result.

######  Result 1 (Nonparametric Equivalence of the Standard DID and Regression
Estimator).

We write the linear regression estimator (equation (A.2)) as a solution to the
following least squares problem.

$(\widehat{\alpha},\widehat{\theta},\widehat{\gamma},\widehat{\beta})=\operatornamewithlimits{argmin}\sum^{n}_{i=1}\sum_{t=1}^{2}O_{it}\Big{\\{}Y_{it}-\alpha-\theta
G_{i}-\gamma I_{t}-\beta(G_{i}\times I_{t})\Big{\\}}^{2}.$

Then, $\widehat{\tau}_{\texttt{DID}}=\widehat{\beta}.$

##### Proof.

By solving the least squares problem, we obtain the following solutions:

$\displaystyle\widehat{\alpha}$ $\displaystyle=\frac{\sum_{i\colon
G_{i}=0}Y_{i1}}{n_{01}}$ $\displaystyle\widehat{\theta}$
$\displaystyle=\frac{\sum_{i\colon G_{i}=1}Y_{i1}}{n_{11}}-\frac{\sum_{i\colon
G_{i}=0}Y_{i1}}{n_{01}}$ $\displaystyle\widehat{\gamma}$
$\displaystyle=\frac{\sum_{i\colon G_{i}=0}Y_{i2}}{n_{02}}-\frac{\sum_{i\colon
G_{i}=0}Y_{i1}}{n_{01}}$ $\displaystyle\widehat{\beta}$
$\displaystyle=\left(\frac{\sum_{i\colon
G_{i}=1}Y_{i2}}{n_{12}}-\frac{\sum_{i\colon
G_{i}=1}Y_{i1}}{n_{11}}\right)-\left(\frac{\sum_{i\colon
G_{i}=0}Y_{i2}}{n_{02}}-\frac{\sum_{i\colon G_{i}=0}Y_{i1}}{n_{01}}\right),$

which completes the proof. ∎

####  C.1.2 Panel Data

Again, for the later use in the Appendix, we report the well-known result that
the standard DID estimator $\widehat{\tau}_{\texttt{DID}}$ (equation (3)) is
equivalent to coefficient $\widehat{\beta}$ in the two-way fixed effects
regression estimator in the panel data setting [1].

######  Result 2 (Nonparametric Equivalence of the Standard DID and Two-way
Fixed Effects Regression Estimator).

We can write the two-way fixed effects regression estimator as a solution to
the following least squares problem.

$(\widehat{\alpha},\widehat{\delta},\widehat{\beta})=\operatornamewithlimits{argmin}\sum^{n}_{i=1}\sum^{2}_{t=1}(Y_{it}-\alpha_{i}-\delta_{t}-\beta
D_{it})^{2}.$

Then, $\widehat{\tau}_{\texttt{DID}}=\widehat{\beta}.$

##### Proof.

First we define the demeaned treatment and outcome variables,
$\overline{Y}_{i}=\sum_{t=1}^{2}Y_{it}/2$,
$\overline{Y}_{t}=\sum_{i=1}^{n}Y_{it}/n$,
$\overline{Y}=\sum_{i=1}^{n}\sum_{t=1}^{2}Y_{it}/2n$,
$\overline{D}_{i}=\sum_{t=1}^{2}D_{it}/2$,
$\overline{D}_{t}=\sum_{i=1}^{n}D_{it}/n$, and
$\overline{D}=\sum_{i=1}^{n}\sum_{t=1}^{2}D_{it}/2n$.

Given these transformed variables, we can transform the least squares problem
into a well-known demeaned form.

$\widehat{\beta}=\operatornamewithlimits{argmin}_{\beta}\sum^{n}_{i=1}\sum^{2}_{t=1}(\widetilde{Y}_{it}-\beta\widetilde{D}_{it})^{2}$

where
$\widetilde{Y}_{it}=Y_{it}-\overline{Y}_{i}-\overline{Y}_{t}+\overline{Y}$ and
$\widetilde{D}_{it}=D_{it}-\overline{D}_{i}-\overline{D}_{t}+\overline{D}$.
Using this notation, we can express $\widehat{\beta}$ as

$\widehat{\beta}=\frac{\sum^{n}_{i=1}\sum^{2}_{t=1}\widetilde{D}_{it}\widetilde{Y}_{it}}{\sum^{n}_{i=1}\sum^{2}_{t=1}\widetilde{D}^{2}_{it}}$

where $\widetilde{D}_{it}$ takes the following form,

$\widetilde{D}_{it}=\begin{cases}1/2\cdot n_{0}/n&\text{ if }G_{i}=1,t=2\\\
-(1/2)\cdot n_{0}/n&\text{ if }G_{i}=1,t=1\\\ -(1/2)\cdot n_{1}/n&\text{ if
}G_{i}=0,t=2\\\ 1/2\cdot n_{1}/n&\text{ if }G_{i}=0,t=1,\end{cases}$

where $n_{1}=\sum_{i=1}^{n}G_{i}$ and $n_{0}=\sum_{i=1}^{n}(1-G_{i})$. Then,
the numerator can be written as

$\displaystyle\sum^{n}_{i=1}\sum^{2}_{t=1}\widetilde{D}_{it}\widetilde{Y}_{it}$
$\displaystyle=\frac{n_{0}}{2n}\bigg{\\{}\sum^{n}_{i=1}G_{i}\widetilde{Y}_{i2}-\sum^{n}_{i=1}G_{i}\widetilde{Y}_{i1}\bigg{\\}}-\frac{n_{1}}{2n}\bigg{\\{}\sum^{n}_{i=1}(1-G_{i})\widetilde{Y}_{i2}-\sum^{n}_{i=1}(1-G_{i})\widetilde{Y}_{i1}\bigg{\\}}$

and the denominator is given as

$\sum^{n}_{i=1}\sum^{2}_{t=1}\widetilde{D}^{2}_{it}=2n_{1}\bigg{(}\frac{n_{0}}{2n}\bigg{)}^{2}+2n_{0}\bigg{(}\frac{n_{1}}{2n}\bigg{)}^{2}=\frac{n_{1}n_{0}}{2n}.$

Combining both terms, we get

$\displaystyle\widehat{\beta}$
$\displaystyle=\frac{\sum^{n}_{i=1}\sum^{2}_{t=1}\widetilde{D}_{it}\widetilde{Y}_{it}}{\sum^{n}_{i=1}\sum^{2}_{t=1}\widetilde{D}^{2}_{it}}$
$\displaystyle=\frac{1}{n_{1}}\bigg{\\{}\sum^{n}_{i=1}G_{i}\widetilde{Y}_{i2}-\sum^{n}_{i=1}G_{i}\widetilde{Y}_{i1}\bigg{\\}}-\frac{1}{n_{0}}\bigg{\\{}\sum^{n}_{i=1}(1-G_{i})\widetilde{Y}_{i2}-\sum^{n}_{i=1}(1-G_{i})\widetilde{Y}_{i1}\bigg{\\}}$
$\displaystyle=\frac{1}{n_{1}}\sum^{n}_{i=1}G_{i}(Y_{i2}-Y_{i1})-\frac{1}{n_{0}}\sum^{n}_{i=1}(1-G_{i})(Y_{i2}-Y_{i1})$
$\displaystyle=\widehat{\tau}_{\texttt{DID}},$

which concludes the proof. ∎

###  C.2 Extended DID

####  C.2.1 Repeated Cross-Sectional Data

We consider a case in which there are two pre-treatment periods $t=\\{0,1\\}$
and one post-treatment period $t=2$. Using this notation, we report the
following result.

######  Result 3 (Nonparametric Equivalence of the Extended DID and Regression
Estimator).

We focus on a linear regression estimator that is a solution to the following
least squares problem.

$(\widehat{\theta},\widehat{\gamma},\widehat{\beta})=\operatornamewithlimits{argmin}\sum^{n}_{i=1}\sum_{t=0}^{2}O_{it}\left(Y_{it}-\theta
G_{i}-\gamma_{t}-\beta D_{it}\right)^{2}.$

Then,
$\widehat{\beta}=\lambda\widehat{\tau}_{\texttt{DID}}+(1-\lambda)\widehat{\tau}_{\texttt{DID(2,0)}}$
where

$\displaystyle\lambda$
$\displaystyle=\cfrac{n_{11}n_{01}(n_{10}+n_{00})}{n_{11}n_{01}(n_{10}+n_{00})+n_{10}n_{00}(n_{11}+n_{01})},$
$\displaystyle 1-\lambda$
$\displaystyle=\cfrac{n_{10}n_{00}(n_{11}+n_{01})}{n_{11}n_{01}(n_{10}+n_{00})+n_{10}n_{00}(n_{11}+n_{01})}.$

When the sample size of each group is fixed over time, i.e., $n_{11}=n_{10}$
and $n_{01}=n_{00}$, $\lambda=1/2$ and therefore, $\widehat{\beta}$ is
equivalent to the extended DID estimator of equal weights in equation (8).

##### Proof.

By solving the least squares problem, we obtain

$\displaystyle\widehat{\theta}$
$\displaystyle=\lambda\left(\frac{\sum_{i\colon
G_{i}=1}Y_{i1}}{n_{11}}-\frac{\sum_{i\colon
G_{i}=0}Y_{i1}}{n_{01}}\right)+(1-\lambda)\left(\frac{\sum_{i\colon
G_{i}=1}Y_{i0}}{n_{10}}-\frac{\sum_{i\colon G_{i}=0}Y_{i0}}{n_{00}}\right)$
$\displaystyle\widehat{\gamma}_{2}$ $\displaystyle=\frac{\sum_{i\colon
G_{i}=0}Y_{i2}}{n_{02}}$ $\displaystyle\widehat{\gamma}_{1}$
$\displaystyle=\frac{\sum_{i\colon G_{i}=1}Y_{i1}+\sum_{i\colon
G_{i}=0}Y_{i1}}{n_{11}+n_{01}}-\frac{n_{11}}{n_{11}+n_{01}}\widehat{\theta}$
$\displaystyle\widehat{\gamma}_{0}$ $\displaystyle=\frac{\sum_{i\colon
G_{i}=1}Y_{i0}+\sum_{i\colon
G_{i}=0}Y_{i0}}{n_{10}+n_{00}}-\frac{n_{10}}{n_{10}+n_{00}}\widehat{\theta}$
$\displaystyle\widehat{\beta}$
$\displaystyle=\lambda\left\\{\left(\frac{\sum_{i\colon
G_{i}=1}Y_{i2}}{n_{12}}-\frac{\sum_{i\colon
G_{i}=1}Y_{i1}}{n_{11}}\right)-\left(\frac{\sum_{i\colon
G_{i}=0}Y_{i2}}{n_{02}}-\frac{\sum_{i\colon
G_{i}=0}Y_{i1}}{n_{01}}\right)\right\\}$
$\displaystyle\qquad+(1-\lambda)\left\\{\left(\frac{\sum_{i\colon
G_{i}=1}Y_{i2}}{n_{12}}-\frac{\sum_{i\colon
G_{i}=1}Y_{i0}}{n_{10}}\right)-\left(\frac{\sum_{i\colon
G_{i}=0}Y_{i2}}{n_{02}}-\frac{\sum_{i\colon
G_{i}=0}Y_{i0}}{n_{00}}\right)\right\\},$

which completes the proof. ∎

####  C.2.2 Panel Data

We report that the extended DID estimator $\widehat{\tau}_{\texttt{e-DID}}$
(equation (8)) (equal weights: $\lambda=1/2$) is equivalent to the estimated
coefficient $\widehat{\beta}$ in the two-way fixed effects regression
estimator in the panel data setting with $t=\\{0,1,2\\}$.

######  Result 4 (Nonparametric Equivalence of the Extended DID and Two-way
Fixed Effects Regression Estimator).

We can write the two-way fixed effects regression estimator as a solution to
the following least squares problem.

$(\widehat{\alpha},\widehat{\delta},\widehat{\beta})=\operatornamewithlimits{argmin}\sum^{n}_{i=1}\sum^{2}_{t=0}(Y_{it}-\alpha_{i}-\delta_{t}-\beta
D_{it})^{2}.$

Then, $\widehat{\tau}_{\texttt{e-DID}}=\widehat{\beta}.$

##### Proof.

First we define $\overline{Y}_{i}=\sum_{t=0}^{2}Y_{it}/3$,
$\overline{Y}_{t}=\sum_{i=1}^{n}Y_{it}/n$,
$\overline{Y}=\sum_{i=1}^{n}\sum_{t=0}^{2}Y_{it}/3n$,
$\overline{D}_{i}=\sum_{t=0}^{2}D_{it}/3$,
$\overline{D}_{t}=\sum_{i=1}^{n}D_{it}/n$, and
$\overline{D}=\sum_{i=1}^{n}\sum_{t=0}^{2}D_{it}/3n.$ Then, we can write the
two-way fixed effects estimator as a two-way demeaned estimator,

$\displaystyle\widehat{\beta}$
$\displaystyle=\operatornamewithlimits{argmin}_{\beta}\sum^{n}_{i=1}\sum^{2}_{t=0}(\widetilde{Y}_{it}-\beta\widetilde{D}_{it})^{2}=\frac{\sum^{n}_{i=1}\sum^{2}_{t=0}\widetilde{D}_{it}\widetilde{Y}_{it}}{\sum^{n}_{i=1}\sum^{2}_{t=0}\widetilde{D}^{2}_{it}},$

as in Result 2, where
$\widetilde{Y}_{it}=Y_{it}-\overline{Y}_{i}-\overline{Y}_{t}+\overline{Y}$ and
$\widetilde{D}_{it}=D_{it}-\overline{D}_{i}-\overline{D}_{t}+\overline{D}$.
Importantly, $\widetilde{D}_{it}$ takes the following form:

$\widetilde{D}_{it}=\begin{cases}2/3\cdot n_{0}/n&\text{ if }G_{i}=1,t=2\\\
-1/3\cdot n_{0}/n&\text{ if }G_{i}=1,t=0,1\\\ -2/3\cdot n_{1}/n&\text{ if
}G_{i}=0,t=2\\\ 1/3\cdot n_{1}/n&\text{ if }G_{i}=0,t=0,1,\end{cases}$

where $n_{1}=\sum_{i=1}^{n}G_{i}$ and $n_{0}=\sum_{i=1}^{n}(1-G_{i})$. Then,
the numerator can be written as

$\displaystyle\sum^{n}_{i=1}\sum^{2}_{t=0}\widetilde{D}_{it}\widetilde{Y}_{it}$
$\displaystyle=\sum^{n}_{i=1}G_{i}\bigg{(}\frac{2n_{0}}{3n}\bigg{)}\widetilde{Y}_{i2}-\sum^{n}_{i=1}\sum_{t=0}^{1}G_{i}\bigg{(}\frac{n_{0}}{3n}\bigg{)}\widetilde{Y}_{it}+\sum^{n}_{i=1}(1-G_{i})\bigg{(}\frac{-2n_{1}}{3n}\bigg{)}\widetilde{Y}_{i2}+\sum^{n}_{i=1}\sum_{t=0}^{1}(1-G_{i})\bigg{(}\frac{n_{1}}{3n}\bigg{)}\widetilde{Y}_{it}$
$\displaystyle=\sum^{n}_{i=1}G_{i}\bigg{(}\frac{n_{0}}{3n}\bigg{)}\\{\widetilde{Y}_{i2}-\widetilde{Y}_{i1}\\}+\sum^{n}_{i=1}G_{i}\bigg{(}\frac{n_{0}}{3n}\bigg{)}\\{\widetilde{Y}_{i2}-\widetilde{Y}_{i0}\\}$
$\displaystyle\qquad-\left\\{\sum^{n}_{i=1}(1-G_{i})\bigg{(}\frac{n_{1}}{3n}\bigg{)}\\{\widetilde{Y}_{i2}-\widetilde{Y}_{i1}\\}+\sum^{n}_{i=1}(1-G_{i})\bigg{(}\frac{n_{1}}{3n}\bigg{)}\\{\widetilde{Y}_{i2}-\widetilde{Y}_{i0}\\}\right\\}$
$\displaystyle=\frac{n_{0}}{3n}\left\\{\sum^{n}_{i=1}G_{i}\\{Y_{i2}-Y_{i1}\\}+\sum^{n}_{i=1}G_{i}\\{Y_{i2}-Y_{i0}\\}\right\\}-\frac{n_{1}}{3n}\left\\{\sum^{n}_{i=1}(1-G_{i})\\{Y_{i2}-Y_{i1}\\}+\sum^{n}_{i=1}(1-G_{i})\\{Y_{i2}-Y_{i0}\\}\right\\}.$

The denominator can be written as

$\displaystyle\sum^{n}_{i=1}\sum^{2}_{t=0}\widetilde{D}^{2}_{it}$
$\displaystyle=\frac{n_{0}n_{1}}{n}\cdot\frac{2}{3}.$

Combining the two terms, we have

$\displaystyle\widehat{\beta}$
$\displaystyle=\frac{1}{2n_{1}}\left\\{\sum^{n}_{i=1}G_{i}\\{Y_{i2}-Y_{i1}\\}+\sum^{n}_{i=1}G_{i}\\{Y_{i2}-Y_{i0}\\}\right\\}$
$\displaystyle\qquad-\frac{1}{2n_{0}}\left\\{\sum^{n}_{i=1}(1-G_{i})\\{Y_{i2}-Y_{i1}\\}+\sum^{n}_{i=1}(1-G_{i})\\{Y_{i2}-Y_{i0}\\}\right\\}$
$\displaystyle=\frac{1}{2}\left\\{\frac{1}{n_{1}}\sum^{n}_{i=1}G_{i}\\{Y_{i2}-Y_{i1}\\}-\frac{1}{n_{0}}\sum^{n}_{i=1}(1-G_{i})\\{Y_{i2}-Y_{i1}\\}\right\\}$
$\displaystyle\qquad+\frac{1}{2}\left\\{\frac{1}{n_{1}}\sum^{n}_{i=1}G_{i}\\{Y_{i2}-Y_{i0}\\}-\frac{1}{n_{0}}\sum^{n}_{i=1}(1-G_{i})\\{Y_{i2}-Y_{i0}\\}\right\\}$
$\displaystyle=\frac{1}{2}\widehat{\tau}_{\texttt{DID}}+\frac{1}{2}\widehat{\tau}_{\texttt{DID(2,0)}}.$

By solving the least squares problem, we also obtain

$\displaystyle\widehat{\alpha}_{i}\ $ $\displaystyle=$ $\displaystyle\
\overline{Y}_{i}-\overline{Y}-\overline{Y}_{t=0}+\widehat{\beta}(\overline{D}-\overline{D}_{t=0})$
$\displaystyle\widehat{\delta}_{t}\ $ $\displaystyle=$ $\displaystyle\
\overline{Y}_{t}-\overline{Y}_{t=0}+\widehat{\beta}(\overline{D}_{t=0}-\overline{D}_{t})$

∎

###  C.3 Sequential DID

The sequential DID estimator is connected to a widely used regression
estimator. In particular, the sequential DID estimator (equation (10)) can be
computed as a linear regression in which we replace the outcome $Y_{it}$ with
a transformed outcome. In panel data, we replace the original outcome with its
first difference $Y_{it}-Y_{i,t-1}$ so that we use changes instead of levels.
In repeated cross-sectional data, we use the following linear regression.

$\Delta
Y_{it}\sim\alpha_{s}+\theta_{s}G_{i}+\gamma_{s}I_{t}+\beta_{s}(G_{i}\times
I_{t}),$ (A.3)

where $\Delta Y_{it}=Y_{it}-(\sum_{i\colon G_{i}=1}Y_{i,t-1})/n_{1,t-1}$ if
$G_{i}=1$ and $\Delta Y_{it}=Y_{it}-(\sum_{i\colon
G_{i}=0}Y_{i,t-1})/n_{0,t-1}$ if $G_{i}=0$. Coefficients are denoted by
$(\alpha_{s},\theta_{s},\gamma_{s},\beta_{s})$. In this case, a coefficient in
front of the interaction term $\beta_{s}$ is numerically identical to the
sequential DID estimator. We provide the proof of this equivalence for both
panel and repeated cross-sectional data settings below.

####  C.3.1 Repeated Cross-Sectional Data

We clarify that the sequential DID estimator $\widehat{\tau}_{\texttt{s-DID}}$
(equation (10)) is equivalent to a coefficient in a regression estimator with
transformed outcomes.

######  Result 5 (Nonparametric Equivalence of the Sequential DID and
Regression Estimator).

We focus on a linear regression estimator with a transformed outcome.

$(\widehat{\alpha},\widehat{\theta},\widehat{\gamma},\widehat{\beta})=\operatornamewithlimits{argmin}\sum^{n}_{i=1}\sum_{t=1}^{2}O_{it}\Big{\\{}\Delta
Y_{it}-\alpha-\theta G_{i}-\gamma I_{t}-\beta(G_{i}\times
I_{t})\Big{\\}}^{2},$

where

$\Delta Y_{it}=\begin{cases}Y_{i2}-\frac{\sum_{i\colon
G_{i}=1}Y_{i1}}{n_{11}}&\text{ if }G_{i}=1,t=2\\\ Y_{i1}-\frac{\sum_{i\colon
G_{i}=1}Y_{i0}}{n_{10}}&\text{ if }G_{i}=1,t=1\\\ Y_{i2}-\frac{\sum_{i\colon
G_{i}=0}Y_{i1}}{n_{01}}&\text{ if }G_{i}=0,t=2\\\ Y_{i1}-\frac{\sum_{i\colon
G_{i}=0}Y_{i0}}{n_{00}}&\text{ if }G_{i}=0,t=1.\end{cases}$

Then, $\widehat{\tau}_{\texttt{s-DID}}=\widehat{\beta}.$

##### Proof.

Using Result 1, we obtain

$\displaystyle\widehat{\beta}$ $\displaystyle=\left(\frac{\sum_{i\colon
G_{i}=1}\Delta Y_{i2}}{n_{12}}-\frac{\sum_{i\colon G_{i}=1}\Delta
Y_{i1}}{n_{11}}\right)-\left(\frac{\sum_{i\colon G_{i}=0}\Delta
Y_{i2}}{n_{02}}-\frac{\sum_{i\colon G_{i}=0}\Delta Y_{i1}}{n_{01}}\right)$
$\displaystyle=\left\\{\left(\frac{\sum_{i\colon
G_{i}=1}Y_{i2}}{n_{12}}-\frac{\sum_{i\colon
G_{i}=1}Y_{i1}}{n_{11}}\right)-\left(\frac{\sum_{i\colon
G_{i}=0}Y_{i2}}{n_{02}}-\frac{\sum_{i\colon
G_{i}=0}Y_{i1}}{n_{01}}\right)\right\\}$
$\displaystyle\qquad\qquad-\left\\{\left(\frac{\sum_{i\colon
G_{i}=1}Y_{i1}}{n_{11}}-\frac{\sum_{i\colon
G_{i}=1}Y_{i0}}{n_{10}}\right)-\left(\frac{\sum_{i\colon
G_{i}=0}Y_{i1}}{n_{01}}-\frac{\sum_{i\colon
G_{i}=0}Y_{i0}}{n_{00}}\right)\right\\},$

which completes the proof. ∎

Next, we clarify that the sequential DID estimator
$\widehat{\tau}_{\texttt{s-DID}}$ (equation (10)) is also equivalent to a
coefficient in a regression estimator with group-specific time trends. [32]
derive similar results by making the parametric assumption of the conditional
expectations. We prove nonparametric equivalence without making any
assumptions about conditional expectations.

######  Result 6 (Nonparametric Equivalence of the Sequential DID and
Regression Estimator with Group-Specific Time Trends).

We focus on a linear regression estimator with group-specific time trends.

$(\widehat{\theta},\widehat{\gamma},\widehat{\beta})=\operatornamewithlimits{argmin}\sum^{n}_{i=1}\sum_{t=0}^{2}O_{it}\Big{\\{}Y_{it}-\theta_{0}G_{i}-\theta_{1}(G_{i}\times
t)-\gamma_{t}-\beta D_{it}\Big{\\}}^{2}.$

Then, $\widehat{\tau}_{\texttt{s-DID}}=\widehat{\beta}.$

##### Proof.

By solving the least squares problem, we obtain

$\displaystyle\widehat{\theta}_{0}$ $\displaystyle=\frac{\sum_{i\colon
G_{i}=1}Y_{i0}}{n_{10}}-\frac{\sum_{i\colon G_{i}=0}Y_{i0}}{n_{00}}$
$\displaystyle\widehat{\theta}_{1}$ $\displaystyle=\left(\frac{\sum_{i\colon
G_{i}=1}Y_{i1}}{n_{11}}-\frac{\sum_{i\colon
G_{i}=0}Y_{i1}}{n_{01}}\right)-\left(\frac{\sum_{i\colon
G_{i}=1}Y_{i0}}{n_{10}}-\frac{\sum_{i\colon G_{i}=0}Y_{i0}}{n_{00}}\right)$
$\displaystyle\widehat{\gamma}_{2}$ $\displaystyle=\frac{\sum_{i\colon
G_{i}=0}Y_{i2}}{n_{02}},\ \ \widehat{\gamma}_{1}=\frac{\sum_{i\colon
G_{i}=0}Y_{i1}}{n_{01}},\ \ \widehat{\gamma}_{0}=\frac{\sum_{i\colon
G_{i}=0}Y_{i0}}{n_{00}}$ $\displaystyle\widehat{\beta}$
$\displaystyle=\left\\{\left(\frac{\sum_{i\colon
G_{i}=1}Y_{i2}}{n_{12}}-\frac{\sum_{i\colon
G_{i}=1}Y_{i1}}{n_{11}}\right)-\left(\frac{\sum_{i\colon
G_{i}=0}Y_{i2}}{n_{02}}-\frac{\sum_{i\colon
G_{i}=0}Y_{i1}}{n_{01}}\right)\right\\}$
$\displaystyle\qquad\qquad-\left\\{\left(\frac{\sum_{i\colon
G_{i}=1}Y_{i1}}{n_{11}}-\frac{\sum_{i\colon
G_{i}=1}Y_{i0}}{n_{10}}\right)-\left(\frac{\sum_{i\colon
G_{i}=0}Y_{i1}}{n_{01}}-\frac{\sum_{i\colon
G_{i}=0}Y_{i0}}{n_{00}}\right)\right\\},$

which completes the proof. ∎

####  C.3.2 Panel Data

We clarify that the sequential DID estimator $\widehat{\tau}_{\texttt{s-DID}}$
(equation (10)) is equivalent to a coefficient in the two-way fixed effects
regression estimator with transformed outcomes.

######  Result 7 (Nonparametric Equivalence of the Sequential DID and Two-way
Fixed Effects Regression Estimator).

We focus on the two-way fixed effects regression estimator with transformed
outcomes.

$(\widehat{\alpha},\widehat{\delta},\widehat{\beta})=\operatornamewithlimits{argmin}\sum^{n}_{i=1}\sum^{2}_{t=1}(\Delta
Y_{it}-\alpha_{i}-\delta_{t}-\beta D_{it})^{2},$

where $\Delta Y_{it}=Y_{it}-Y_{i,t-1}$. Then,
$\widehat{\tau}_{\texttt{s-DID}}=\widehat{\beta}.$

##### Proof.

As in Result 2, we can focus on the demeaned form.

$\widehat{\beta}=\operatornamewithlimits{argmin}\sum^{n}_{i=1}\sum^{2}_{t=1}(\widetilde{\Delta
Y}_{it}-\beta\widetilde{D}_{it})^{2},$

where $\widetilde{\Delta Y}_{it}=\Delta Y_{it}-\overline{\Delta
Y}_{i}-\overline{\Delta Y}_{t}+\overline{\Delta Y}$, $\overline{\Delta
Y}_{i}=\sum_{t=1}^{2}\Delta Y_{it}/2$, $\overline{\Delta
Y}_{t}=\sum_{i=1}^{n}\Delta Y_{it}/n$, and $\overline{\Delta
Y}=\sum_{i=1}^{n}\sum_{t=1}^{2}\Delta Y_{it}/2n$. Similarly,
$\widetilde{D}_{it}=D_{it}-\overline{D}_{i}-\overline{D}_{t}+\overline{D}$,
$\overline{D}_{i}=\sum_{t=1}^{2}D_{it}/2$,
$\overline{D}_{t}=\sum_{i=1}^{n}D_{it}/n$, and
$\overline{D}=\sum_{i=1}^{n}\sum_{t=1}^{2}D_{it}/2n.$ Using Result 2,

$\displaystyle\widehat{\beta}$
$\displaystyle=\frac{1}{n_{1}}\sum^{n}_{i=1}G_{i}(\Delta Y_{i2}-\Delta
Y_{i1})-\frac{1}{n_{0}}\sum^{n}_{i=1}(1-G_{i})(\Delta Y_{i2}-\Delta Y_{i1})$
$\displaystyle=\bigg{\\{}\frac{1}{n_{1}}\sum^{n}_{i=1}G_{i}(Y_{i2}-Y_{i1})-\frac{1}{n_{0}}\sum^{n}_{i=1}(1-G_{i})(Y_{i2}-Y_{i1})\bigg{\\}}$
$\displaystyle\qquad\quad-\bigg{\\{}\frac{1}{n_{1}}\sum^{n}_{i=1}G_{i}(Y_{i1}-Y_{i0})-\frac{1}{n_{0}}\sum^{n}_{i=1}(1-G_{i})(Y_{i1}-Y_{i0})\bigg{\\}}$
$\displaystyle\equiv\widehat{\tau}_{\texttt{s-DID}},$

which concludes the proof. ∎

Next, we clarify that the sequential DID estimator
$\widehat{\tau}_{\texttt{s-DID}}$ (equation (10)) is also equivalent to a
coefficient in the two-way fixed effects regression estimator with individual-
specific time trends.

######  Result 8 (Nonparametric Equivalence of the Sequential DID and Two-way
Fixed Effects Regression Estimator with Individual-Specific Time Trends).

We focus on the two-way fixed effects regression estimator with individual-
specific time trends

$(\widehat{\alpha},\widehat{\xi},\widehat{\delta},\widehat{\beta})=\operatornamewithlimits{argmin}\sum^{n}_{i=1}\sum^{2}_{t=0}(Y_{it}-\alpha_{i}-(\xi_{i}\times
t)-\delta_{t}-\beta D_{it})^{2}.$

Then, $\widehat{\tau}_{\texttt{s-DID}}=\widehat{\beta}.$

##### Proof.

By solving the least squares problem, we obtain that

$\displaystyle\sum_{\mathclap{i\colon G_{i}=1}}Y_{i2}$
$\displaystyle=(\widehat{\beta}+\widehat{\gamma}_{2})n_{1}+\sum_{\mathclap{i\colon
G_{i}=1}}\widehat{\alpha}_{i}+2\sum_{\mathclap{i\colon
G_{i}=1}}\widehat{\xi}_{i},\quad\sum_{\mathclap{i\colon
G_{i}=0}}Y_{i2}=\widehat{\gamma}_{2}n_{0}+\sum_{\mathclap{i\colon
G_{i}=0}}\widehat{\alpha}_{i}+2\sum_{\mathclap{i\colon
G_{i}=0}}\widehat{\xi}_{i}$ $\displaystyle\sum_{\mathclap{i\colon
G_{i}=1}}Y_{i1}$
$\displaystyle=\widehat{\gamma}_{1}n_{1}+\sum_{\mathclap{i\colon
G_{i}=1}}\widehat{\alpha}_{i}+\sum_{\mathclap{i\colon
G_{i}=1}}\widehat{\xi}_{i},\quad\qquad\quad\ \sum_{\mathclap{i\colon
G_{i}=0}}Y_{i1}=\widehat{\gamma}_{1}n_{0}+\sum_{\mathclap{i\colon
G_{i}=0}}\widehat{\alpha}_{i}+\sum_{\mathclap{i\colon
G_{i}=0}}\widehat{\xi}_{i}$ $\displaystyle\sum_{\mathclap{i\colon
G_{i}=1}}Y_{i0}$
$\displaystyle=\widehat{\gamma}_{0}n_{1}+\sum_{\mathclap{i\colon
G_{i}=1}}\widehat{\alpha}_{i},\quad\qquad\qquad\qquad\quad\sum_{\mathclap{i\colon
G_{i}=0}}Y_{i0}=\widehat{\gamma}_{0}n_{0}+\sum_{\mathclap{i\colon
G_{i}=0}}\widehat{\alpha}_{i}.$

Therefore, we get

$\displaystyle\widehat{\beta}$
$\displaystyle=\left\\{\left(\frac{\sum_{i\colon
G_{i}=1}Y_{i2}}{n_{1}}-\frac{\sum_{i\colon
G_{i}=1}Y_{i1}}{n_{1}}\right)-\left(\frac{\sum_{i\colon
G_{i}=0}Y_{i2}}{n_{0}}-\frac{\sum_{i\colon
G_{i}=0}Y_{i1}}{n_{0}}\right)\right\\}$
$\displaystyle\qquad\qquad-\left\\{\left(\frac{\sum_{i\colon
G_{i}=1}Y_{i1}}{n_{1}}-\frac{\sum_{i\colon
G_{i}=1}Y_{i0}}{n_{1}}\right)-\left(\frac{\sum_{i\colon
G_{i}=0}Y_{i1}}{n_{0}}-\frac{\sum_{i\colon
G_{i}=0}Y_{i0}}{n_{0}}\right)\right\\},$

which completes the proof. ∎

####  C.3.3 Alternative Interpretation of Parallel Trends-in-Trends Assumption

We emphasize an alternative way to interpret the parallel trends-in-trends
assumption. Unlike the parallel trends assumption that assumes the time-
invariant unmeasured confounding, the parallel trends-in-trends assumption can
account for linear time-varying unmeasured confounding — unobserved
confounding increases or decreases over time but with some constant rate. For
example, researchers might be worried that some treated communes have higher
motivation for reforms, which is not measured, and the infrastructure
qualities differ between treated and control communes due to this unobserved
motivation. The parallel trends assumption means that the difference in the
infrastructure qualities due to this unobserved confounder does not grow or
decline over time. In contrast, the parallel trends-in-trends assumption
accommodates a simple yet important case in which the unobserved difference in
the infrastructure qualities does grow or decline with some fixed rate, which
analysts do not need to specify. This interpretation comes from the following
equivalent representation of the parallel trends-in-trends assumption.

$\displaystyle\underbrace{\\{\mathbb{E}[Y_{i2}(0)\mid
G_{i}=1]-\mathbb{E}[Y_{i2}(0)\mid G_{i}=0]\\}}_{\text{Bias at
$t=2$}}-\underbrace{\\{\mathbb{E}[Y_{i1}(0)\mid
G_{i}=1]-\mathbb{E}[Y_{i1}(0)\mid G_{i}=0]\\}}_{\text{ Bias at $t=1$}}$
$\displaystyle=\underbrace{\\{\mathbb{E}[Y_{i1}(0)\mid
G_{i}=1]-\mathbb{E}[Y_{i1}(0)\mid G_{i}=0]\\}}_{\text{ Bias at
$t=1$}}-\underbrace{\\{\mathbb{E}[Y_{i0}(0)\mid
G_{i}=1]-\mathbb{E}[Y_{i0}(0)\mid G_{i}=0]\\}}_{\text{ Bias at $t=0$}}.$ (A.4)

The difference between the mean potential outcome $Y_{it}(0)$ for the treated
and control group at time $t$, $\mathbb{E}[Y_{it}(0)\mid
G_{i}=1]-\mathbb{E}[Y_{it}(0)\mid G_{i}=0]$, is often called _bias_ (or
selection bias) in the literature [21, 12, e.g.,]. Equation (A.4) shows that
the parallel trends-in-trends assumption allows for a linear change in bias
over time, whereas the bias is assumed to be constant over time in the
extended parallel trends assumption. This representation is useful when we
generalize our results to $K$ pre-treatment periods where $K>2$. Importantly,
equation (11) and equation (A.4) are equivalent, and therefore, researchers
can choose whichever interpretation easy for them to evaluate in each
application.

###  C.4 Connection to the Leads Test

Here we formally prove the connection between the test of pre-treatment
periods discussed in Section 2.2 and the well known leads test [3]. The leads
test includes $D_{i,t+1}$ into a linear regression and check whether a
coefficient of $D_{i,t+1}$ is zero.

####  C.4.1 Repeated Cross-Sectional Data

In the repeated cross-sectional data setting, the leads test considers the
following linear regression.

$(\widehat{\theta},\widehat{\gamma},\widehat{\beta},\widehat{\zeta})=\operatornamewithlimits{argmin}\sum^{n}_{i=1}\sum_{t=0}^{1}O_{it}\left(Y_{it}-\theta
G_{i}-\gamma_{t}-\beta D_{it}-\zeta D_{i,t+1}\right)^{2}.$

Then, because $D_{it}=0$ for all units in $t=\\{0,1\\}$, this least squares
problem is the same as

$(\widehat{\theta},\widehat{\gamma},\widehat{\zeta})=\operatornamewithlimits{argmin}\sum^{n}_{i=1}\sum_{t=0}^{1}O_{it}\left(Y_{it}-\theta
G_{i}-\gamma_{t}-\zeta D_{i,t+1}\right)^{2}.$

Finally, using Result 1, we have

$\displaystyle\widehat{\zeta}$ $\displaystyle=\left(\frac{\sum_{i\colon
G_{i}=1}Y_{i1}}{n_{11}}-\frac{\sum_{i\colon
G_{i}=1}Y_{i0}}{n_{10}}\right)-\left(\frac{\sum_{i\colon
G_{i}=0}Y_{i1}}{n_{01}}-\frac{\sum_{i\colon G_{i}=0}Y_{i0}}{n_{00}}\right),$

which is the standard DID estimator to the pre-treatment periods $t=0,1.$ ∎

####  C.4.2 Panel Data

In the panel data setting, the leads test considers the following two-way
fixed effects regression.

$(\widehat{\alpha},\widehat{\delta},\widehat{\beta},\widehat{\zeta})=\operatornamewithlimits{argmin}\sum^{n}_{i=1}\sum_{t=0}^{1}(Y_{it}-\alpha_{i}-\delta_{t}-\beta
D_{it}-\zeta D_{i,t+1})^{2}.$

Again, this least squares problem is the same as

$(\widehat{\alpha},\widehat{\delta},\widehat{\zeta})=\operatornamewithlimits{argmin}\sum^{n}_{i=1}\sum_{t=0}^{1}(Y_{it}-\alpha_{i}-\delta_{t}-\zeta
D_{i,t+1})^{2}.$

Then, using Result 2, we have

$\displaystyle\widehat{\zeta}$ $\displaystyle=\left(\frac{\sum_{i\colon
G_{i}=1}Y_{i1}}{n_{1}}-\frac{\sum_{i\colon
G_{i}=1}Y_{i0}}{n_{1}}\right)-\left(\frac{\sum_{i\colon
G_{i}=0}Y_{i1}}{n_{0}}-\frac{\sum_{i\colon G_{i}=0}Y_{i0}}{n_{0}}\right),$

which is the standard DID estimator to the pre-treatment periods $t=0,1.$ ∎

##  Appendix D Details of Double DID Estimator

###  D.1 Properties of Double DID Estimator

Here, we prove several important properties of the double DID estimator based
on the GMM theory [18].

######  Theorem 1.

When the extended parallel trends assumption (Assumption 2) holds, the double
DID estimator with the optimal weight matrix (equation (13) in the main paper)
is consistent, and its asymptotic variance is smaller than or equal to that of
the standard, extended, and sequential DID estimators, i.e., ${\rm
Var}(\widehat{\tau}_{\texttt{d-DID}})\leq\mbox{min}({\rm
Var}(\widehat{\tau}_{\texttt{DID}}),{\rm
Var}(\widehat{\tau}_{\texttt{s-DID}}),{\rm
Var}(\widehat{\tau}_{\texttt{e-DID}})).$

### Proof.

Suppose we define a moment function $m_{i}(\tau)$ as

$m_{i}(\tau)=\begin{pmatrix}\tau-\widehat{\tau}_{\texttt{DID}}(i)\\\\[-5.0pt]
\tau-\widehat{\tau}_{\texttt{s-DID}}(i)\end{pmatrix}$

where

$\displaystyle\widehat{\tau}_{\texttt{DID}}(i)$ $\displaystyle=$
$\displaystyle\left(\frac{n}{n_{12}}G_{i}Y_{i2}-\frac{n}{n_{11}}G_{i}Y_{i1}\right)-\left(\frac{n}{n_{02}}(1-G_{i})Y_{i2}-\frac{n}{n_{01}}(1-G_{i})Y_{i1}\right)$
$\displaystyle\widehat{\tau}_{\texttt{s-DID}}(i)$ $\displaystyle=$
$\displaystyle\left\\{\left(\frac{n}{n_{12}}G_{i}Y_{i2}-\frac{n}{n_{11}}G_{i}Y_{i1}\right)-\left(\frac{n}{n_{02}}(1-G_{i})Y_{i2}-\frac{n}{n_{01}}(1-G_{i})Y_{i1}\right)\right\\}$
$\displaystyle\hskip
14.45377pt-\left\\{\left(\frac{n}{n_{11}}G_{i}Y_{i1}-\frac{n}{n_{10}}G_{i}Y_{i0}\right)-\left(\frac{n}{n_{01}}(1-G_{i})Y_{i1}-\frac{n}{n_{00}}(1-G_{i})Y_{i0}\right)\right\\}$

for the repeated cross-sectional setting. They can be similarly defined in the
panel data setting. Then, we can write the double DID estimator as the GMM
estimator:

$\widehat{\tau}_{\texttt{d-DID}}(\mathbf{W})=\operatornamewithlimits{argmin}_{\tau}\left(\frac{1}{n}\sum_{i=1}^{n}m_{i}(\tau)\right)^{\top}\mathbf{W}\left(\frac{1}{n}\sum_{i=1}^{n}m_{i}(\tau)\right)$
(A.2)

where we index the double DID estimator by $\mathbf{W}$, which is a weight
matrix of dimension $2\times 2$.

In general, the variance of the GMM estimator is given by

${\rm
Var}(\widehat{\tau}_{\texttt{d-DID}}(\mathbf{W}))=(M^{\top}\mathbf{W}M)^{-1}M^{\top}\mathbf{W}\Omega\mathbf{W}^{\top}M(M^{\top}\mathbf{W}M)^{-1}.$

where
$M=\frac{1}{n}\sum_{i=1}^{n}\mathbb{E}\left\\{\frac{\partial}{\partial\tau}m_{i}(\tau)\right\\},$
and

$\Omega=\begin{pmatrix}{\rm
Var}(\widehat{\tau}_{\texttt{DID}})&\mbox{Cov}(\widehat{\tau}_{\texttt{DID}},\widehat{\tau}_{\texttt{s-DID}})\\\
\mbox{Cov}(\widehat{\tau}_{\texttt{DID}},\widehat{\tau}_{\texttt{s-DID}})&{\rm
Var}(\widehat{\tau}_{\texttt{s-DID}})\end{pmatrix}.$

[18] showed in general that ${\rm
Var}(\widehat{\tau}_{\texttt{d-DID}}(\mathbf{W}))$ is minimized when
$\mathbf{W}$ is set to $\Omega^{-1}.$ We define this optimal weight as
$\mathbf{W}^{\ast}$

$\mathbf{W}^{\ast}=\Omega^{-1}=\begin{pmatrix}{\rm
Var}(\widehat{\tau}_{\texttt{DID}})&\mbox{Cov}(\widehat{\tau}_{\texttt{DID}},\widehat{\tau}_{\texttt{s-DID}})\\\
\mbox{Cov}(\widehat{\tau}_{\texttt{DID}},\widehat{\tau}_{\texttt{s-DID}})&{\rm
Var}(\widehat{\tau}_{\texttt{s-DID}})\end{pmatrix}^{-1}.$

In general, the asymptotic variance of this optimal GMM estimator is given by

${\rm
Var}(\widehat{\tau}_{\texttt{d-DID}}(\mathbf{W}^{\ast}))=(M^{\top}\mathbf{W}^{\ast}M)^{-1}.$

Because $M=\bm{1},$ the asymptotic variance of ${\rm
Var}(\widehat{\tau}_{\texttt{d-DID}}(\mathbf{W}^{\ast}))$ can be explicitly
written as

${\rm
Var}(\widehat{\tau}_{\texttt{d-DID}}(\mathbf{W}^{\ast}))=(\bm{1}^{\top}\mathbf{W}^{\ast}\bm{1})^{-1}=\cfrac{{\rm
Var}(\widehat{\tau}_{\texttt{DID}})\cdot{\rm
Var}(\widehat{\tau}_{\texttt{s-DID}})-\mbox{Cov}(\widehat{\tau}_{\texttt{DID}},\widehat{\tau}_{\texttt{s-DID}})^{2}}{{\rm
Var}(\widehat{\tau}_{\texttt{DID}})+{\rm
Var}(\widehat{\tau}_{\texttt{s-DID}})-2\mbox{Cov}(\widehat{\tau}_{\texttt{DID}},\widehat{\tau}_{\texttt{s-DID}})}.$

Finally, the standard, sequential, and extended DID estimators are all special
cases of the double DID with a specific choice of the weight matrix as
described in Table 1 of the main paper. Because for any $\mathbf{W},$ ${\rm
Var}(\widehat{\tau}_{\texttt{d-DID}}(\mathbf{W}^{\ast}))\leq{\rm
Var}(\widehat{\tau}_{\texttt{d-DID}}(\mathbf{W}))$, it implies that

${\rm
Var}(\widehat{\tau}_{\texttt{d-DID}}(\mathbf{W}^{\ast}))\leq\mbox{min}({\rm
Var}(\widehat{\tau}_{\texttt{DID}}),{\rm
Var}(\widehat{\tau}_{\texttt{s-DID}}),{\rm
Var}(\widehat{\tau}_{\texttt{e-DID}})).$

Now, we can show the consistency of the estimator and its variance estimator.
The optimal weight matrix $\mathbf{W}^{\ast}$ can be estimated by its sample
analog:

$\widehat{\mathbf{W}}=\begin{pmatrix}\widehat{{\rm
Var}}(\widehat{\tau}_{\texttt{DID}})&\widehat{\mbox{Cov}}(\widehat{\tau}_{\texttt{DID}},\widehat{\tau}_{\texttt{s-DID}})\\\
\widehat{\mbox{Cov}}(\widehat{\tau}_{\texttt{DID}},\widehat{\tau}_{\texttt{s-DID}})&\widehat{{\rm
Var}}(\widehat{\tau}_{\texttt{s-DID}})\end{pmatrix}^{-1}.$

which is a consistent estimator of $\mathbf{W}^{*}$ under the standard
regularity conditions. Therefore, by solving the GMM optimization problem
(equation (A.2)), we can explicitly write the double DID as

$\widehat{\tau}_{\texttt{d-DID}}(\widehat{\mathbf{W}})=\widehat{w}_{1}\widehat{\tau}_{\texttt{DID}}+\widehat{w}_{2}\widehat{\tau}_{\texttt{s-DID}}$

where $\widehat{w}_{1}+\widehat{w}_{2}=1$, and

$\displaystyle\widehat{w}_{1}=\cfrac{\widehat{{\rm
Var}}(\widehat{\tau}_{\texttt{s-DID}})-\widehat{\mbox{Cov}}(\widehat{\tau}_{\texttt{DID}},\widehat{\tau}_{\texttt{s-DID}})}{\widehat{{\rm
Var}}(\widehat{\tau}_{\texttt{DID}})+\widehat{{\rm
Var}}(\widehat{\tau}_{\texttt{s-DID}})-2\widehat{\mbox{Cov}}(\widehat{\tau}_{\texttt{DID}},\widehat{\tau}_{\texttt{s-DID}})},$
$\displaystyle\widehat{w}_{2}=\cfrac{\widehat{{\rm
Var}}(\widehat{\tau}_{\texttt{DID}})-\widehat{\mbox{Cov}}(\widehat{\tau}_{\texttt{DID}},\widehat{\tau}_{\texttt{s-DID}})}{\widehat{{\rm
Var}}(\widehat{\tau}_{\texttt{DID}})+\widehat{{\rm
Var}}(\widehat{\tau}_{\texttt{s-DID}})-2\widehat{\mbox{Cov}}(\widehat{\tau}_{\texttt{DID}},\widehat{\tau}_{\texttt{s-DID}})}.$

Under the extended parallel trends assumption (Assumption 2), both the
standard DID and the sequential DID estimator are consistent to the ATT.
Therefore, by the continuous mapping theorem and law of large numbers, we have

$\widehat{\tau}_{\texttt{d-DID}}(\widehat{\mathbf{W}})\overset{p}{\to}\tau$

and

$\widehat{{\rm
Var}}(\widehat{\tau}_{\texttt{d-DID}}(\widehat{\mathbf{W}}))\overset{p}{\to}{\rm
Var}(\widehat{\tau}_{\texttt{d-DID}}(\mathbf{W}^{*})),$

which complets the proof. ∎

###  D.2 Standard Error Estimation

As described in Section 3.1.2, we use the block bootstrap.

  1. 1.

Estimate
$\\{\widehat{\tau}^{(b)}_{\texttt{DID}},\widehat{\tau}^{(b)}_{\texttt{s-DID}}\\}^{B}_{b=1}$
where $B$ indicates the total number of bootstrap iterations. We recommend the
block-bootstrap where the block is taken at the level of treatment assignment.

  2. 2.

Estimate the optimal weight matrix via computing the variance-covariance
matrix:

$\displaystyle\widehat{\text{Var}}(\widehat{\tau}_{\texttt{DID}})$
$\displaystyle=\frac{1}{B}\sum^{B}_{b=1}(\widehat{\tau}^{(b)}_{\texttt{DID}}-\overline{\widehat{\tau}}_{\texttt{DID}})^{2}$
$\displaystyle\widehat{\text{Var}}(\widehat{\tau}_{\texttt{s-DID}})$
$\displaystyle=\frac{1}{B}\sum^{B}_{b=1}(\widehat{\tau}^{(b)}_{\texttt{s-DID}}-\overline{\widehat{\tau}}_{\texttt{s-DID}})^{2}$
$\displaystyle\widehat{\text{Cov}}(\widehat{\tau}_{\texttt{DID}},\widehat{\tau}_{\texttt{s-DID}})$
$\displaystyle=\frac{1}{B}\sum^{B}_{b=1}(\widehat{\tau}^{(b)}_{\texttt{DID}}-\overline{\widehat{\tau}}_{\texttt{DID}})(\widehat{\tau}^{(b)}_{\texttt{s-DID}}-\overline{\widehat{\tau}}_{\texttt{s-DID}})$

where
$\overline{\widehat{\tau}}_{\texttt{DID}}=\sum^{B}_{b=1}\widehat{\tau}^{(b)}_{\texttt{DID}}/B$,
and
$\overline{\widehat{\tau}}_{\texttt{s-DID}}=\sum^{B}_{b=1}\widehat{\tau}^{(b)}_{\texttt{s-DID}}/B$
are empirical average of two estimators. Finally, we obtain the estimate of
the weight matrix by inverting the variance-covariance matrix (equation (13)
in the main text),

$\widehat{\mathbf{W}}=\left(\begin{array}[]{cc}\widehat{\text{Var}}(\widehat{\tau}_{\texttt{DID}})&\widehat{\text{Cov}}(\widehat{\tau}_{\texttt{DID}},\widehat{\tau}_{\texttt{s-DID}})\\\
\widehat{\text{Cov}}(\widehat{\tau}_{\texttt{DID}},\widehat{\tau}_{\texttt{s-DID}})&\widehat{\text{Var}}(\widehat{\tau}_{\texttt{s-DID}})\end{array}\right)^{-1}$

  3. 3.

The double DID estimator is given by equation (14) in the main paper.

  4. 4.

The variance of double DID estimator is then obtained via the standard
efficient GMM variance formula

$\widehat{\text{Var}}(\widehat{\tau}_{\texttt{d-DID}})=(\bm{1}^{\top}\widehat{\mathbf{W}}\bm{1})^{-1}.$

##  Appendix E Extensions of Double DID

###  E.1 Double DID Regression

Like other DID estimators, the double DID estimator has a nice connection to a
widely-used regression approach. Using this double DID regression, researchers
can include other pre-treatment covariates $\mathbf{X}_{it}$ to make the DID
design more robust and efficient. We provide technical details in Appendix

To introduce the regression-based double DID estimator, we begin with the
basic DID. As discussed in Appendix C.1, the basic DID estimator is equivalent
to a coefficient in the linear regression of equation (A.2). Inspired by this
connection, researchers often adjust for additional pre-treatment covariates
as:

$Y_{it}\sim\alpha+\theta G_{i}+\gamma I_{t}+\beta(G_{i}\times
I_{t})+\mathbf{X}_{it}^{\top}\bm{\rho},$ (A.3)

where we adjust for the additional pre-treatment covariates $\mathbf{X}_{it}$.
A coefficient of the interaction term $\widehat{\beta}$ is a consistent
estimator for the ATT when the conditional parallel trends assumption holds
and the parametric model is correctly specified. Here, we make the parallel
trends assumption conditional on covariates $\mathbf{X}_{it}$. The idea is
that even when the parallel trends assumption might not hold without
controlling for any covariates, trends of the two groups might be parallel
conditionally after adjusting for observed covariates. For example, the
conditional parallel trends assumption means that treatment and control groups
have the same trends of the infrastructure quality after controlling for
population and GDP per capita.

The sequential DID estimator is extended similarly. Based on the connection to
the linear regression of equation (A.3), we can adjust for additional pre-
treatment covariates as:

$\Delta
Y_{it}\sim\alpha_{s}+\theta_{s}G_{i}+\gamma_{s}I_{t}+\beta_{s}(G_{i}\times
I_{t})+\mathbf{X}_{it}^{\top}\bm{\rho}_{s},$ (A.4)

where $\Delta Y_{it}=Y_{it}-(\sum_{i\colon G_{i}=1}Y_{i,t-1})/n_{1,t-1}$ if
$G_{i}=1$ and $\Delta Y_{it}=Y_{it}-(\sum_{i\colon
G_{i}=0}Y_{i,t-1})/n_{0,t-1}$ if $G_{i}=0$. The estimated coefficient
$\widehat{\beta}_{s}$ is consistent for the ATT under the conditional parallel
trends-in-trends assumption and the conventional assumption of correct
specification.

The double DID regression combines the two regression estimators via the GMM:

$\widehat{\beta}_{\texttt{d-DID}}=\operatornamewithlimits{argmin}_{\beta_{d}}\begin{pmatrix}\beta_{d}-\widehat{\beta}\\\\[-5.0pt]
\beta_{d}-\widehat{\beta}_{s}\end{pmatrix}^{\top}\mathbf{W}\begin{pmatrix}\beta_{d}-\widehat{\beta}\\\\[-5.0pt]
\beta_{d}-\widehat{\beta}_{s}\end{pmatrix}$ (A.5)

where $\mathbf{W}$ is a weighting matrix of dimension $2\times 2$.

Thus, as the double DID estimator with no covariates, the double DID
regression has two steps. The first step is to assess the underlying
assumptions. Here, instead of using the standard DID estimator, we use the
standard DID regression on pre-treatment periods to assess the conditional
extended parallel trends assumption. The second step is to estimate the ATT,
while adjusting for pre-treatment covariates. Instead of using the double DID
estimator without covariates, we implement the regression-based double DID
estimator (equation (A.5)).

###  E.2 Generalized $K$-DID

In this section, we propose the generalized $K$-DID, which extends the double
DID in Section 3 to arbitrary number of pre\- and post-treatment periods in
the basic DID setting. We consider the staggered adoption design in Section 4.

####  E.2.1 The Setup and Causal Quantities of Interest

We first extend the setup to account for arbitrary number of pre- and post-
treatment periods. Suppose we observe outcome $Y_{it}$ for
$i\in\\{1,\ldots,n\\}$ and $t\in\\{0,1,\ldots,T\\}$. We define the binary
treatment variable to be $D_{it}\in\\{0,1\\}$. The treatment is assigned right
before time period $T^{\ast}$, and thus, time periods
$t\in\\{T^{\ast},\ldots,T\\}$ are the post-treatment periods and time periods
$t\in\\{0,\ldots,T^{\ast}-1\\}$ are the pre-treatment periods. As in Section
2, we denote the treatment group as $G_{i}=1$ and $G_{i}=0$ otherwise. Note
that $D_{it}=0$ for $t\in\\{1,\ldots,T^{\ast}\\}$ for all units.

We are interested in the causal effect at post-treatment time $T^{\ast}+s$
where $s\geq 0$. When $s=0$, this corresponds to the contemporaneous treatment
effect. By specifying different values of $s>0,$ researchers can study a
variety of long-term causal effects of the treatment. Formally, our quantity
of interest is the average treatment effect on the treated (ATT) at post-
treatment time $T^{\ast}+s$.

$\tau(s)\equiv\mathbb{E}[Y_{i,T^{\ast}+s}(1)-Y_{i,T^{\ast}+s}(0)\mid
G_{i}=1].$

For example, when $s=3$, this could mean the causal effect of the policy after
three years from its initial introduction. This definition is a generalization
of the standard ATT: when $s=0,$ this quantity is equal to the ATT defined in
equation (1).

####  E.2.2 Generalize Parallel Trends Assumptions

What assumptions do we need to identify the ATT at post-treatment time
$T^{\ast}+s$? Here, we provide a generalization of the parallel trends
assumption, which incorporates both the standard parallel trends assumption
and the parallel trends-in-trends assumption.

######  Assumption A1 ($k$-th Order Parallel Trends).

For some integer $k$ such that $1\leq k\leq T^{\ast},$

$\Delta_{s}^{k}\left(\mathbb{E}[Y_{i,T^{\ast}+s}(0)\mid
G_{i}=1]\right)=\Delta_{s}^{k}\left(\mathbb{E}[Y_{i,T^{\ast}+s}(0)\mid
G_{i}=0]\right),$

where $\Delta_{s}^{k}$ is the $k$-th order difference operator defined
recursively as follows. For $g\in\\{0,1\\}$,

$\displaystyle\Delta^{1}_{s}\left(\mathbb{E}[Y_{i,T^{\ast}+s}(0)\mid
G_{i}=g]\right)\equiv\mathbb{E}[Y_{i,T^{\ast}+s}(0)\mid
G_{i}=g]-\mathbb{E}[Y_{i,T^{\ast}-1}(0)\mid G_{i}=g],$

when $k=1$ and, in general,

$\displaystyle\Delta_{s}^{k}\left(\mathbb{E}[Y_{i,T^{\ast}+s}(0)\mid
G_{i}=g]\right)$ $\displaystyle\equiv\ $
$\displaystyle\Delta^{k-1}_{s}\left(\mathbb{E}[Y_{i,T^{\ast}+s}(0)\mid
G_{i}=g]\right)-M^{k}_{s}\Delta^{k-1}\left(\mathbb{E}[Y_{i,T^{\ast}-1}(0)\mid
G_{i}=g]\right),$ $\displaystyle=\ $
$\displaystyle\mathbb{E}[Y_{i,T^{\ast}+s}(0)\mid
G_{i}=g]-\mathbb{E}[Y_{i,T^{\ast}-1}(0)\mid
G_{i}=g]-\sum_{j=1}^{k-1}M^{j+1}_{s}\Delta^{j}\left(\mathbb{E}[Y_{i,T^{\ast}-1}(0)\mid
G_{i}=g]\right),$

where $M^{\ell}_{s}=\prod_{j=1}^{\ell-1}(s+j)/\prod_{j=1}^{\ell-1}j$ for
$\ell\geq 2$. $\Delta^{k}\left(\mathbb{E}[Y_{i,T^{\ast}-1}(0)\mid
G_{i}=g]\right)$ is also recursively defined as
$\Delta^{k}\left(\mathbb{E}[Y_{i,T^{\ast}-1}(0)\mid
G_{i}=g]\right)\equiv\Delta^{k-1}\left(\mathbb{E}[Y_{i,T^{\ast}-1}(0)\mid
G_{i}=g]\right)-\Delta^{k-1}\left(\mathbb{E}[Y_{i,T^{\ast}-2}(0)\mid
G_{i}=g]\right),$ and $\Delta^{1}\left(\mathbb{E}[Y_{i,T^{\ast}-m}(0)\mid
G_{i}=g]\right)=\mathbb{E}[Y_{i,T^{\ast}-m}(0)\mid
G_{i}=g]-\mathbb{E}[Y_{i,T^{\ast}-m-1}(0)\mid G_{i}=g]$ for $m=\\{1,2\\}.$ The
standard parallel trends assumption and the parallel-trends-in-trends
assumption are both special cases of this assumption. The $k$-th order
parallel trends assumption reduces to the standard parallel trends assumption
(Assumption 1) when $s=1$ and $k=1$, and to the parallel-trends-in-trends
assumption (Assumption 3) when $s=1$ and $k=2.$

To further clarify the meaning of Assumption A1, we can consider a simpler but
stronger condition. In particular, the $k$-th order parallel trends assumption
(Assumption A1) is implied by the following $p$-th degree polynomial model of
confounding.

$\mathbb{E}[Y_{it}(0)\mid G_{i}=1]-\mathbb{E}[Y_{it}(0)\mid
G_{i}=0]=\alpha+\sum_{p=1}^{k-1}\Gamma_{p}t^{p},$

with unknown parameters $\alpha$ and $\bm{\Gamma}$. Here, the left hand side
of the equality captures the difference between the two groups (treatment and
control) in terms of the mean of potential outcomes under the control
condition. This representation shows that the standard parallel trends
assumption (Assumption 1) is implied by the time-invariant confounding; the
parallel trends-in-trends assumption (Assumption 3) is implied by the linear
time-varying confounding; and in general, the $k$-th order parallel trends
assumption is implied by the $k$-th order polynomial confounding.

####  E.2.3 Estimate ATT with Multiple Pre- and Post-Treatment Periods

We consider the identification and estimation of the ATT at post-treatment
time $T^{\ast}+s$. Under the $k$-th order parallel trends assumption
(Assumption A1), the ATT is identified as follows.

$\tau(s)=\Delta_{s}^{k}\left(\mathbb{E}[Y_{i,T^{\ast}+s}\mid
G_{i}=1]\right)-\Delta_{s}^{k}\left(\mathbb{E}[Y_{i,T^{\ast}+s}\mid
G_{i}=0]\right).$

Because each conditional expectation can be consistently estimated via its
sample analogue,

$\widehat{\tau}_{k}(s)=\Delta_{s}^{k}\left(\frac{\sum_{i\colon
G_{i}=1}Y_{i,T^{\ast}+s}}{n_{1,T^{\ast}+s}}\right)-\Delta_{s}^{k}\left(\frac{\sum_{i\colon
G_{i}=0}Y_{i,T^{\ast}+s}}{n_{0,T^{\ast}+s}}\right)$

is a consistent estimator for the ATT at time $T^{\ast}+s$ under the $k$-th
order parallel trends assumption. When $s=0$ and $k=1$, this estimator
corresponds to the standard DID estimator (equation (3)). When $s=0$ and
$k=2,$ this is equal to the sequential DID estimator (equation (10)). While
existing approaches [3, 33, 28, 32, e.g.,] consider each estimator separately,
we propose combining multiple DID estimators within the GMM framework.

In general, the generalized double DID combines $K$ moment conditions where
$K$ is the number of pre-treatment periods researchers use. When there are
more than two pre-treatment periods, we can naturally combine more than two
DID estimators, improving upon the double DID in Section 3. Formally, the
generalized double DID is defined as,

$\widehat{\tau}(s)=\operatornamewithlimits{argmin}_{\tau}\bm{g}(\tau)^{\top}\widehat{\mathbf{W}}\bm{g}(\tau)$

where
$\bm{g}(\tau)=(\tau-\widehat{\tau}_{1}(s),\ldots,\tau-\widehat{\tau}_{K}(s))^{\top}$.
Based on the theory of the efficient GMM [18], the optimal weight matrix is
$\widehat{\mathbf{W}}={\rm Var}(\widehat{\tau}_{(1:K)}(s))^{-1}$ where ${\rm
Var}(\cdot)$ is the variance-covariance matrix and
$\widehat{\tau}_{(1:K)}(s)=(\widehat{\tau}_{1}(s),\ldots,\widehat{\tau}_{K}(s))^{\top}.$
When $T^{\ast}=2,$ this converges to the standard DID estimator (equation
(3)). When $T^{\ast}=3,$ this corresponds to the basic form of the double DID
estimator (equation (12)). Within the GMM framework, we can select moment
conditions using the J-statistics [18]. We can similarly generalize the double
DID regression.

To assess the extended parallel trends assumption, we can apply the
generalized double DID to pre-treatment periods
$t\in\\{1,\ldots,T^{\ast}-1\\}$ as if the last pre-treatment period
$T^{\ast}-1$ is the target time period. Moments are
$\bm{g}(\tau)=(\tau-\widehat{\tau}_{1}(0),\ldots,\tau-\widehat{\tau}_{K}(0))^{\top}$
where $\widehat{\tau}_{k}(0)=\Delta_{s}^{k}\left(\frac{\sum_{i\colon
G_{i}=1}Y_{i,T^{\ast}-1}}{n_{1,T^{\ast}-1}}\right)-\Delta_{s}^{k}\left(\frac{\sum_{i\colon
G_{i}=0}Y_{i,T^{\ast}-1}}{n_{0,T^{\ast}-1}}\right).$ Similarly, to assess the
extended parallel trends-in-trends assumption, we can apply the generalized
double DID to pre-treatment periods with moments
$\bm{g}(\tau)=(\tau-\widehat{\tau}_{2}(0),\ldots,\tau-\widehat{\tau}_{K}(0))^{\top}$.

###  E.3 Generalized $K$-DID for Staggered Adoption Design

Combining the setup introduced in Section E.2.1 and the one in Section 4.1, we
propose the generalized $K$-DID for the SA design, which allows researchers to
estimate long-term causal effects in the SA design. We focus on the SA-ATT at
post-treatment time $t+s$ where $t$ is the timing of the treatment assignment
and $s\geq 0$ represents how far in the future we want estimate the ATT for.
We first redefine the group indicator $G$ to estimate the long-term SA-ATT at
post-treatment time $t+s$. In particular, we define

$G_{its}\ =\ \begin{cases}\ \ 1&\mbox{if }A_{i}=t\\\ \ \ 0&\mbox{if
}A_{i}>t+s\\\ \ \ -1&\mbox{otherwise}\end{cases}$

where $G_{its}=1$ represents units who receive the treatment at time $t$, and
$G_{its}=0$ indicates units who do not receive the treatment by time $t+s$.
$G_{its}=-1$ includes other units who receive the treatment before time $t$ or
receive the treatment between $t+1$ and $t+s.$ When $s=0$, this definition
corresponds to the group indicator in equation (15).

Formally, our first quantity of interest is the staggered-adoption average
treatment effect on the treated (SA-ATT) at post-treatment time $t+s$.

$\tau^{\textsf{SA}}(s,t)\equiv\mathbb{E}[Y_{i,t+s}(1)-Y_{i,t+s}(0)\mid
G_{its}=1].$

By averaging over time, we can also define the time-average staggered-adoption
average treatment effect on the treated (time-average SA-ATT) at $s$ periods
after treatment onset.

$\overline{\tau}^{\textsf{SA}}(s)\equiv\sum_{t\in\mathcal{T}}\pi_{t}\tau^{\textsf{SA}}(s,t),$

where $\mathcal{T}$ represents a set of the time periods for which researchers
want to estimate the ATT. The SA-ATT in period $t$, $\tau^{\textsf{SA}}(t),$