The exact contents of citations.db main_text.text for this paper — one flattened LaTeX string, title through conclusion, appendix excluded, unmodified except for removing email addresses. This is what our citation measures are computed over.
117,289 characters
Difference-in-Differences Meets Synthetic Control: Doubly Robust Identification and Estimation
\title{Difference-in-Differences Meets Synthetic Control: Doubly Robust
Identification and Estimation\thanks{
Authors are listed in alphabetical order; all contributed equally to this
work. We gratefully acknowledge Yawei Wang for outstanding research
assistance.}}
\author{Yixiao Sun\thanks{
University of California, San Diego. Email: \url{[email removed]}.} \and
Haitian Xie\thanks{
Corresponding author. Peking University. Email: \url{[email removed]}.}
\and Yuhang Zhang\thanks{
The Chinese University of Hong Kong. Email: \url{[email removed]}.}}
\date{\today}
\maketitle
\vspace{-2.5em}
\begin{abstract}
\singlespacing Difference-in-Differences (DiD) and Synthetic Control (SC) are widely used methods for causal inference in panel data, each with distinct strengths and limitations. We propose a novel method for short-panel causal inference that integrates the advantages of both approaches. Our method delivers a doubly robust identification strategy for the average treatment effect on the treated (ATT) under either of two non-nested assumptions: parallel trends or a group-level SC condition. Building on this identification result, we develop a unified semiparametric framework for estimating the ATT. Notably, the identification-robust moment function satisfies Neyman orthogonality under the parallel trends assumption but not under the SC assumption, leading to different asymptotic variances across the two identification strategies. To ensure valid inference, we propose a multiplier bootstrap method that consistently approximates the asymptotic distribution under either assumption. Furthermore, we extend our methodology to accommodate repeated cross-sectional data and staggered treatment designs. As an empirical application, we evaluate the impact of the 2003 minimum wage increase in Alaska on family income. Finally, in simulation studies based on empirically calibrated data-generating processes, we demonstrate that the proposed estimation and inference methods perform well in finite samples under either identification assumption.
\bigskip \noindent \textbf{Keywords}: Difference-in-Differences, Doubly
Robust Identification, Multiplier Bootstrap, Semiparametric Estimation,
Synthetic Control.
\bigskip \noindent \textbf{JEL codes:} C14, C21, C23
\end{abstract}
\vspace{-3em}
\vspace{-2em}
\newpage
\section{Introduction}
In recent years, Difference-in-Differences (DiD) methods have gained
significant traction in economics and the social sciences, playing a central
role in the \textquotedblleft credibility revolution\textquotedblright\
\citep{AngristPischke2010_credibility}. Over 30\% of NBER applied
microeconomics working papers in 2024 mention DiD or related event study
methods, surpassing alternatives such as instrumental variables and
regression discontinuity designs \citep{goldsmith2024tracking}. Yet, DiD's
reliance on the parallel trends assumption, a requirement often scrutinized
as an oversimplification of real-world dynamics, leaves researchers
vulnerable to biased estimates when trends diverge.
The synthetic control method --- \textquotedblleft arguably the most
important innovation in the policy-evaluation literature in the last 15
years\textquotedblright\ \citep{athey2017state} --- constructs
counterfactuals by matching treated units to weighted combinations of
untreated units. It has surged in popularity in settings where the
parallel-trends assumption may fail. However, its use is now largely
confined to studies with long panels of aggregate data, such as countries,
states, or regions
\citep{abadie2003economic, abadie2010synthetic, hsiao2012panel,
abadie2015comparative}, rather than to short panels of micro-level data with
rich individual information. Given the limited availability of micro-level
datasets with long time horizons, the method has not yet fulfilled its full
potential in applied microeconomic research.
In this paper, we address these limitations by adopting a different
perspective on the synthetic-control problem and integrating the strengths
of DiD and synthetic control methods. We focus on a typical microeconomic
panel data setting where individuals are observed repeatedly over time and
grouped into aggregate-level units (such as households within states), with
treatment assigned at the group level. We introduce a novel causal inference
method that unifies the identification strategies of DiD and synthetic
control in a doubly robust framework. Our approach nonparametrically
identifies the average treatment effect on the treated (ATT) under either
the DiD parallel trends assumption or the synthetic control assumption. This
robustness enables applied researchers to avoid the conventional trade-off
between DiD and synthetic control, thereby strengthening the credibility of
causal inference.
To operationalize our method, we propose a semiparametric estimation
procedure and a bootstrap inference approach.\footnote{
We use \textquotedblleft semiparametric\textquotedblright\ to describe a
setting in which the parameters of interest are finite-dimensional, while
nuisance parameters, such as the propensity score, conditional outcome
expectations, and synthetic control weights, are specified nonparametrically.
} Developing asymptotic theory within this framework presents substantial
challenges. Under the parallel trends assumption, the proposed moment
condition that identifies the ATT satisfies Neyman orthogonality, allowing
flexible estimation of nuisance parameters \citep{chernozhukov2018DML}.
However, under the synthetic control structure, the moment condition does
not satisfy Neyman orthogonality, requiring careful adjustment to the
asymptotic variance to account for the estimated nuisance parameters.
Importantly, the asymptotic variance differs depending on the identification
assumption. To ensure that the inference procedure remains doubly robust, we
propose a multiplier bootstrap method that consistently approximates the
asymptotic distribution under either assumption. Our approach achieves
double robustness across three key dimensions: identification, estimation,
and inference, ensuring both consistent point estimation and valid
statistical inference under either identifying assumption.
We further extend the method to accommodate two additional settings:
repeated cross-sectional data and staggered treatment designs. For repeated
cross-sectional data, different sets of individuals are sampled across
different time periods. We show that the doubly robust identification
results from the panel setting extend naturally to the repeated
cross-sectional setting, provided that the variables are stationary over
time, a standard assumption in this context. For staggered treatment
designs, we build on the recent DiD literature
\citep[e.g.,][]{callaway2021difference}, using untreated groups as controls
for the treatment group of interest. This approach effectively reduces the
problem to a setting with a single treatment group and a single treatment
period.
In our empirical study, we apply the proposed method to evaluate the effect of Alaska’s 2003 minimum wage increase on family income, using the Current Population Survey with states as groups and households as units. Consistent with \citet{gunsilius2023distributional}, we find no statistically significant immediate impact of the policy on family income.
In the simulation study, we calibrate the data-generating processes to match the distribution observed in the empirical application. We consider three DGPs: (i) only the parallel trends assumption holds, (ii) only the synthetic control assumption holds, and (iii) both assumptions hold. The proposed estimation and inference procedures demonstrate consistently good performance across all three DGPs.
We organize the remainder of the paper as follows. The rest of this section
discusses the relevant literature. Section \ref{sec:setup} introduces the
panel model setup, along with the identification assumptions and the doubly
robust identification result. Section \ref{sec:asymptotics} establishes the
semiparametric estimation theory and the multiplier bootstrap theory
separately under the parallel trends and synthetic control assumptions.
Sections \ref{sec:rc} and \ref{sec:staggered} extend the analysis to
repeated cross-sectional data and staggered treatment designs, respectively.
Sections \ref{sec:empirical} and \ref{sec:simulation} present the empirical and simulation studies. Section \ref
{sec:conclusion} concludes the paper. Proofs of the theoretical results are
provided in the appendix.
\paragraph{Related literature}
Our paper contributes to three strands of literature. First, it contributes
to the modern semiparametric panel data and DiD literature
\citep{heckman1997matching,abadie2005semiparametric, sant2020doubly, callaway2021difference, chen2025efficient}. When the parallel trends
assumption holds, our estimator can be interpreted as a weighted average of
doubly robust DiD estimators across different groups, inheriting their
robustness to parametric misspecification of nuisance parameters and their
compatibility with double/debiased machine learning (DML) frameworks
\citep{chernozhukov2018DML}. More importantly, our method introduces a novel
dimension of robustness: even when parallel trends fail due to unobserved
confounding, our estimator remains valid if the synthetic control structure
holds. This innovation addresses concerns about the fragility of parallel
trends in applied work, offering researchers a safeguard against violations
of this key assumption.
Second, our paper contributes to the growing literature on synthetic control
methods, particularly in the context of micro-level data with a short time
dimension. While classical synthetic control approaches have predominantly
focused on aggregate-level data, recent studies have begun to explore
applications at the individual level. For example, relying on factor models,
\cite{robbins2017framework} examined the impact of region-level crime
interventions using individual-level data, whereas \cite
{chen2020distributional} introduced a distributional synthetic control
method based on a quantile factor model. More recently, nonparametric
approaches, such as those proposed by \cite{gunsilius2023distributional} and
\cite{chen2023group}, have been developed to study the synthetic matching of
the entire outcome distribution by drawing on the changes-in-changes method
from \cite{athey2006identification}. While \cite{gunsilius2023distributional}
's method applies to short-panel settings, \cite{chen2023group} requires a
large number of pre-treatment periods. Complementing these developments, our
approach integrates synthetic control methods within the modern causal
inference framework, requiring only a short time dimension. It achieves
nonparametric identification and semiparametric estimation results that
align with those in recent DiD and other causal methods, thereby broadening
the scope of synthetic control applications in empirical research.
Third, our work extends the literature on causal panel data methods, such as the matrix completion method \citep{athey2021matrix},
the synthetic DiD method \citep{arkhangelsky2021synthetic}, and the augmented SC method \citep{ben2021augmented}. These papers operate on aggregate-level data and, to drive bias to zero, typically assume a large number of control groups and/or a large number of pre-treatment periods. In contrast, we work with short panels of individual-level data organized into heterogeneous groups. Our framework formalizes the synthetic control structure as a nonparametric identification assumption, with covariates entering naturally as conditioning variables.
A prototypical example of such group structures is the classification of individuals or households by their state of residence, a common practice in state-level policy studies that rely on widely used microeconomic datasets, such as the Current Population Survey,
American Community Survey, Panel Study of Income Dynamics, and National
Longitudinal Surveys. By leveraging this group structure with granular information, we can identify the synthetic control weights without relying on either a large number of pre-treatment periods or a
large number of aggregate control groups. Instead, we assume that the number
of individuals in each group is large, and thus our asymptotic framework is
different. A more detailed discussion is provided in Section \ref
{sec:no-covariate}.
\section{Setup and identification}
\label{sec:setup}
\subsection{Panel model with a single treated group}
We initiate our investigation with a panel data model, where we observe
repeated outcomes over time for the same individuals. Our data consists of a
collection of groups, which represent aggregate-level units, such as
countries, states, provinces, cities, or other similar entities. Within each
aggregate-level unit, there are individual units. The categorical variable $
G $ signifies the specific group to which an individual unit belongs. Denote
$N_{G}$ as the total number of control groups. The first group, $g=1$, is
the treated group, and the remaining groups, $g=2,\ldots ,N_{G}+1,$ are the
control groups. These units are observed over $\mathcal{T}$ time periods
denoted by $t\in \{1,\ldots ,\mathcal{T}\}$.\footnote{
We do not use the usual notation $T$ here because it denotes the time stamp
for individual units in the repeated cross-sectional setting in Section \ref
{sec:rc}.} For a group $g$, let $\mathcal{G}_{g}\equiv \mathbf{1}\{G=g\}$ be
the binary variable (0 or 1) that indicates whether an individual unit
belongs to group $g.$
A binary, irreversible treatment is assigned to the groups. In periods $
t=1,\dots,\mathcal{T}-1$, all groups remain untreated. In the final period $
t=\mathcal{T}$, the first group receives treatment, while the other groups
remain untreated. We focus on this scenario with a single treated group and
a single treatment period, as it allows us to more effectively illustrate
the methodological contribution in this simpler setting. An extension to
staggered treatment designs, involving multiple treatment groups and
variation in treatment timing, is presented in Section \ref{sec:staggered}.
The potential outcomes are defined over the entire treatment trajectory. Let
$\bm{0}_{s}$ denote a vector of zeros of dimension $s$. We define $\tilde{Y}
_{t}(\bm{0}_{\mathcal{T}-1},1)$ as the potential outcome at period $t$ if
the individual receives treatment in the final period and no treatment in
all preceding periods. Similarly, we define $\tilde{Y}_{t}(\bm{0}_{\mathcal{T
}-1},0)\equiv \tilde{Y}_{t}(\bm{0}_{\mathcal{T}})$ as the potential outcome
if the individual never receives treatment. This definition of potential
outcomes will be useful when extending the framework to account for
staggered treatment adoption.
In the scenario of a single treatment time, for simplicity and with a slight
abuse of notation, we use $Y_{t}(1)$ and $Y_{t}(0)$ to represent $\tilde{Y}
_{t}(\bm{0}_{\mathcal{T}-1},1)$ and $\tilde{Y}_{t}(\bm{0}_{\mathcal{T}-1},0)$
, respectively. The observed outcome at time $t$ is given by $Y_{t}\equiv
\mathcal{G}_{1}Y_{t}(1)+(1-\mathcal{G}_{1})Y_{t}(0)$.
\begin{comment}
\begin{namedassumption} {NA}[No-Anticipation] \label{asm:NA}
For any pretreatment period $t \in\{1,\ldots,\mathcal{T}-1\}$, $Y_t(1) = Y_t(0)$.
\end{namedassumption}
\end{comment}
Under the no-anticipation assumption, an individual's observed pre-treatment
outcome is equal to the untreated potential outcome. Therefore, the observed
outcome is given by
\begin{align*}
Y_{t}& =Y_{t}(0),t=1,\ldots ,\mathcal{T}-1, \\
Y_{\mathcal{T}}& =\mathcal{G}_{1}Y_{\mathcal{T}}(1)+(1-\mathcal{G}_{1})Y_{
\mathcal{T}}(0).
\end{align*}
The target parameter is
\begin{equation*}
{\Greekmath 0112} \equiv \mathbb{E}[Y_{\mathcal{T}}(1)-Y_{\mathcal{T}}(0)|G=1],
\end{equation*}
which represents the average treatment effect for the treated (ATT) in the
post-treatment period. For simplicity, we focus on identifying and
estimating this mean parameter, although this approach can be easily
extended to, for example, the distributional change from $Y_{\mathcal{T}}(0)$
to $Y_{\mathcal{T}}(1)$ in the treatment group. This can be achieved by
considering $\mathbb{E}\left[ h(Y_{\mathcal{T}}(1))\mid G=1\right] -\mathbb{E
}\left[ h(Y_{\mathcal{T}}(0))\mid G=1\right] $ for an appropriately chosen
function $h$.
For each individual unit, let $X$ represent a set of time-invariant
covariates that describe the individual's characteristics. Denote the
support of $X$ by $\mathcal{X}$.
\begin{comment}
\begin{namedassumption}{PDS}[Panel Data Sampling] \label{asm:panel}
We observe an independent and identically distributed (iid) sample $\{S_i \equiv (Y_{1i},\ldots,Y_{\mathcal{T}i},G_i,X_i): 1 \leq i \leq n\}$ of $S \equiv (Y_{1},\ldots,Y_{\mathcal{T}},G,X)$.
\end{namedassumption}
\end{comment}
Under Assumption \ref{asm:panel}, we have $n$ individual-level units indexed
by $i=1,\ldots,n$. The number of such units in group $g$ is the cardinality
of the set $\{i:G_{i}=g\}=\{i:\mathcal{G}_{gi}=1\}$. \
\subsection{Identification assumptions}
We introduce the following three identification assumptions, with the
understanding that they need not all be imposed at once.
\begin{comment}
\begin{namedassumption}{O}[Overlap] \label{asm:overlap}
For all groups $g=1,\ldots,N_G+1$, the propensity score $p_g(X) \equiv \mathbb{P}(G=g|X) > 0$ almost surely.
\end{namedassumption}
\end{comment}
\begin{comment}
\begin{namedassumption}{PT}[Parallel Trends] \label{asm:PT}
For each control group $g \geq 2$, $\mathbb{E}[Y_{\mathcal{T}}(0) - Y_{\mathcal{T}-1}(0)|G=1,X] = \mathbb{E}[Y_{\mathcal{T}}(0) - Y_{\mathcal{T}-1}(0)|G=g,X]$ almost surely.
\end{namedassumption}
\end{comment}
\begin{comment}
\begin{namedassumption}{SC}[Synthetic Control] \label{asm:SC}
There exists a vector of weight functions $w \equiv (w_2, \ldots, w_{N_G+1})$ for which the elements sum to one and satisfy that for almost all $x \in \mathcal{X}$,
\begin{align} \label{eqn:w-id}
\mathbb{E}[Y_{t}(0)|G=1,X=x] = \sum_{g=2}^{N_G+1} w_g(x) \mathbb{E}[Y_{t}(0)|G=g,X=x], t=1,\ldots,\mathcal{T}.
\end{align}
Furthermore, $\mathcal{T} \geq N_G$.\footnote{Theoretically, imposing nonnegativity on the weights is not necessary in our framework, although such a restriction is commonly maintained in the classical synthetic control literature \citep{abadie2021using}. If the true weights are nonnegative, this structure can be imposed on the weight estimation without altering the asymptotic theory for the proposed ATT estimator.}
\end{namedassumption}
\end{comment}
Assumption \ref{asm:PT} adopts a nonparametric parallel trends condition
that fits naturally within our framework. First, it is common practice in
empirical research to estimate panel regressions on individual-level data
with group-level (e.g., state- or county-level) fixed effects and common
time trends. Such models inherently assume that groups share the same trend
while allowing for group-specific fixed effects. Second, concerns about the
plausibility of parallel trends typically arise in settings with long
pre-treatment windows. In contrast, our assumption applies to only a single
period, requiring a common trend solely between the last pre-treatment
period and the treatment period. Compared with the two-way fixed-effects
model, which presumes a common trend over the entire horizon, this is a
markedly weaker temporal restriction. Third, when there are more than two
control groups, the parallel trends assumption allows for overidentification
tests, enabling researchers to empirically assess whether the control groups
follow a parallel trend. Control groups that deviate from this trend can be
excluded based on these tests. Of course, this approach requires identifying
at least one control group that satisfies the parallel trends assumption, a
choice that must rely on domain knowledge rather than data. Finally, the
assumption is made conditional on the covariate $X$. Therefore, we only need
to maintain the parallel trends assumption for the subset of the treatment
group and the corresponding subset of the control group with the same
covariate value. This is more plausible than an unconditional parallel
trends assumption, especially when the covariates that may affect the
outcome variable differ between the treatment and control groups
\citep{abadie2005semiparametric}.
Assumption \ref{asm:SC} is the group-level representation of the synthetic
control structure. It requires that, for each covariate value, the
conditional mean of the outcome in the treated group can be represented as a
linear combination of the conditional means for the untreated groups. We
allow the weights to depend on covariates, which accommodates the
possibility that the weighting patterns vary across units with different
covariate values. In other words, the synthetic control structure is imposed
only on subsets of the treatment and control groups that share the same
covariate value, enhancing the assumption's plausibility. We further assume
that the weighting structure remains the same over time, allowing us to
identify the weighting scheme using the pre-treatment data and apply it in
the post-treatment period to identify the ATT. For the weights to be
identified, it is necessary that $\mathcal{T}\geq N_{G}$.
One can also motivate Assumption \ref{asm:SC} with a factor model structure,
as in the conventional synthetic control literature:
\begin{equation*}
Y_{i,t}(0)={\Greekmath 0115} (G_{i},X_{i})^{\prime }F_{t}+{\Greekmath 0122} _{i,t},\qquad
\mathbb{E}[{\Greekmath 0122} _{i,t}|X_{i},G_{i}]=0,
\end{equation*}
where $F_{t}$ is a vector of common factors, and ${\Greekmath 0115}(G_i,X_i)$ is a vector of factor loadings. Then Assumption \ref{asm:SC} holds if there exist weighting
functions $w_{g}\left( \cdot \right) $ such that
\begin{equation*}
{\Greekmath 0115} (1,x)=\sum_{g=2}^{N_{G}+1}w_{g}\left( x\right) {\Greekmath 0115} \left(
g,x\right),
\end{equation*}
for almost all $x\in \mathcal{X}.$ Because we impose no parametric form on
the weighting functions $w_{g}\left( \cdot \right) $, the factor loading $
{\Greekmath 0115} (G_{i},X_{i})$ may vary across individuals through their group
membership and observed covariates, and there is no parametric restriction
on ${\Greekmath 0115} (\cdot ,\cdot ).$
The structure in Assumption \ref{asm:SC} takes a different approach from the
distributional synthetic control framework in \cite
{gunsilius2023distributional}. Specifically, we focus on matching the
expectation of the outcome rather than the entire distribution, and we allow
the weights to depend on covariates, while \cite{gunsilius2023distributional}
does not. We require that the number of time periods ($\mathcal{T}$) be at
least as large as the number of control groups ($N_{G}$) to identify the
weights. This differs from the traditional synthetic control method, which
relies on an increasing number of pre-treatment periods for consistent
estimation. Here, we only require the number of time periods to exceed the
number of control groups, without assuming that it diverges to infinity.
\footnote{
If the number of time periods is smaller than the number of control groups,
i.e., $\mathcal{T}<N_{G}$, a natural and practical approach is to leverage
prior studies or domain knowledge to retain only the most relevant control
groups for comparison with the treatment group. An alternative approach is
to incorporate a penalty term in the weight estimation process, similar to
the synthetic DiD method \citep{arkhangelsky2021synthetic}, to enable
estimation even when the weights are not point-identified. However,
incorporating such penalization into semiparametric estimation adds further
complexity, which we leave for future research.} This condition is often met
in practice, such as in monthly datasets like the Current Population Survey,
where states serve as the groups. Unlike the canonical synthetic control
method, our framework does not require an increasing number of time periods
because it leverages a growing number of individual units within each group.
An alternative identification assumption commonly used in the synthetic
control literature is the conditional independence of post-treatment $
Y_{t}(0)$ and treatment assignment, given the pre-treatment potential
outcomes
\citep[e.g.,][]{robbins2017framework,ding2019bracketing,kellogg2021combining}
. This condition, often referred to as ignorability conditional on lagged
outcomes, is relatively strong. In contrast, the parallel trends condition
is conceptually less stringent while still preserving the dynamic nature of
the model.
In summary, Assumptions \ref{asm:PT} and \ref{asm:SC} reflect two different
approaches to identification. Assumption \ref{asm:PT} relies on a strong
cross-sectional relationship among groups, though this relationship needs
only to hold over a single period. In contrast, Assumption \ref{asm:SC}
allows for a weaker cross-sectional relationship but requires it to persist
over a longer duration. Importantly, neither Assumption \ref{asm:PT} implies
Assumption \ref{asm:SC}, nor vice versa; these two assumptions are
nonnested. In practice, it is challenging for researchers to decide which
assumption to rely on, as each leads to different estimation methods and
possibly very different estimates.
\subsection{Doubly robust identification}
We introduce the identification results for the ATT. Let $p\equiv
(p_{g}\left( \cdot \right) ,g=1,\ldots ,N_{G}+1)$ and $w\equiv (w_{g}\left(
\cdot \right) ,g=2,\ldots ,N_{G}+1)$. For simplicity, denote $\Delta Y\equiv
\Delta Y_{\mathcal{T}}\equiv Y_{\mathcal{T}}-Y_{\mathcal{T}-1}$. Let $
m_{\Delta }(X)\equiv \mathbb{E}[Y_{\mathcal{T}}-Y_{\mathcal{T}-1}|G\neq 1,X]$
, and ${\Greekmath 0119} _{1}\equiv \mathbb{P}(G=1)$. Note that ${\Greekmath 0119} _{1}$ is strictly
positive, as implied by Assumption \ref{asm:overlap}. Our identification
strategy is to construct a moment function whose mean is the target causal
parameter ${\Greekmath 0112} $. We propose the following moment function:
\begin{equation}
{\Greekmath 011E} (S;m_{\Delta },p,w;{\Greekmath 0119} _{1})\equiv \frac{\mathcal{G}_{1}}{{\Greekmath 0119} _{1}}
(\Delta Y-m_{\Delta }(X))-\frac{1}{{\Greekmath 0119} _{1}}\sum_{g=2}^{N_{G}+1}w_{g}(X)
\frac{p_{1}(X)}{p_{g}(X)}\mathcal{G}_{g}(\Delta Y-m_{\Delta }(X)),
\label{eqn:phi-expression1}
\end{equation}
which can be equivalently represented as
\begin{equation}
{\Greekmath 011E} (S;m_{\Delta },p,w;{\Greekmath 0119} _{1})=\frac{1}{{\Greekmath 0119} _{1}}\left( \mathcal{G}
_{1}-\sum_{g=2}^{N_{G}+1}w_{g}(X)\frac{p_{1}(X)}{p_{g}(X)}\mathcal{G}
_{g}\right) (\Delta Y-m_{\Delta }(X)). \label{eqn:phi-expression2}
\end{equation}
Before showing that ${\Greekmath 011E} \left( \cdot \right) $ is unbiased for ${\Greekmath 0112}$ under either
Assumption \ref{asm:PT} or Assumption \ref{asm:SC}, we provide some
intuition for its construction. First, if we assume only the parallel trends
condition, the standard approach is to estimate the ATT using the DiD
estimand $\mathbb{E}[\mathcal{G}_{1}(\Delta Y-m_{\Delta }(X))/{\Greekmath 0119} _{1}]$. In
this case, equation (\ref{eqn:phi-expression1}) reveals that ${\Greekmath 011E} $ can be
viewed as the DiD formula augmented with a synthetic-control adjustment
term. Conversely, if we assume only the synthetic control condition, the
estimand for the ATT would be
\begin{equation*}
\mathbb{E}\left[ \frac{1}{{\Greekmath 0119} _{1}}\left( \mathcal{G}_{1}-
\sum_{g=2}^{N_{G}+1}w_{g}(X)\frac{p_{1}(X)}{p_{g}(X)}\mathcal{G}_{g}\right)
Y_{\mathcal{T}}\right] .
\end{equation*}
Here, (\ref{eqn:phi-expression2}) shows that ${\Greekmath 011E} $ represents the
synthetic control formula with a DiD-style adjustment, replacing $Y_{\mathcal{T}}$ with $(\Delta Y-m_{\Delta }(X))$.
We can gain further insight into the moment function by considering a
special case. Specifically, in the absence of covariate $X,$ the mean
function $m_{\Delta }$ becomes
\begin{equation*}
m_{\Delta }=\mathbb{E}\left[ Y_{\mathcal{T}}-Y_{\mathcal{T}-1}|G\neq 1\right]
,
\end{equation*}
which is the expected change in the outcome of interest from time $\mathcal{T
}-1$ to $\mathcal{T}$ for the control groups. Then
\begin{align*}
\frac{1}{{\Greekmath 0119} _{1}}\mathbb{E}\left[ \mathcal{G}_{1}\left( \Delta Y-m_{\Delta
}\right) \right] & =\mathbb{E}\left[ \left( \Delta Y-m_{\Delta }\right) |G=1
\right] =\mathbb{E}\left[ \Delta Y|G=1\right] -\mathbb{E}\left[ m_{\Delta
}|G=1\right] \\
& =\mathbb{E}\left[ Y_{\mathcal{T}}-Y_{\mathcal{T}-1}|G=1\right] -\mathbb{E}
\left[ Y_{\mathcal{T}}-Y_{\mathcal{T}-1}|G\neq 1\right] ,
\end{align*}
which takes the familiar DiD form. On the other hand, in the absence of $X,$
$w_{g}$ and $p_{g}$ become constants, and $p_{1}={\Greekmath 0119} _{1}.$ Then
\begin{align*}
\frac{1}{{\Greekmath 0119} _{1}}\mathbb{E}\left[ \left( \mathcal{G}_{1}-
\sum_{g=2}^{N_{G}+1}w_{g}\frac{p_{1}}{p_{g}}\mathcal{G}_{g}\right) Y_{
\mathcal{T}}\right] & =\frac{1}{{\Greekmath 0119} _{1}}\mathbb{E}\left[ \mathcal{G}_{1}Y_{
\mathcal{T}}\right] -\frac{1}{{\Greekmath 0119} _{1}}\mathbb{E}\sum_{g=2}^{N_{G}+1}w_{g}
\frac{p_{1}}{p_{g}}\mathcal{G}_{g}Y_{\mathcal{T}} \\
& =\mathbb{E}\left[ Y_{\mathcal{T}}|\mathcal{G}_{1}=1\right]
-\sum_{g=2}^{N_{G}+1}w_{g}\mathbb{E}\left[ Y_{\mathcal{T}}|\mathcal{G}_{g}=1
\right] \\
& =\mathbb{E}\left[ Y_{\mathcal{T}}|G=1\right] -\sum_{g=2}^{N_{G}+1}w_{g}
\mathbb{E}\left[ Y_{\mathcal{T}}|G=g\right] ,
\end{align*}
which is the familiar synthetic control formula. These calculations make it
clear that ${\Greekmath 011E} $ is equal to a DiD-based moment function with a synthetic
control adjustment, or, equivalently, ${\Greekmath 011E} $ is equal to a synthetic
control-based moment function with a DiD-type adjustment.
In general, ${\Greekmath 011E} $ integrates elements from both approaches and remains
unbiased under either of the identification assumptions, as established in
the following theorem.
\begin{theorem}
\label{thm:identification} Let Assumptions \ref{asm:NA}, \ref{asm:panel},
and \ref{asm:overlap} hold. \
\begin{enumerate}
\item If Assumption \ref{asm:PT} holds, then
\begin{align*}
{\Greekmath 0112} & = \mathbb{E}[{\Greekmath 011E}(S;m_{\Delta},\tilde{p},\tilde{w};{\Greekmath 0119}_1)] =
\mathbb{E}[{\Greekmath 011E}(S;\tilde{m}_{\Delta},p,\tilde{w};{\Greekmath 0119}_1)],
\end{align*}
for any set of weights $\Tilde{w}$ that sum to one, any non-zero functions $
\Tilde{p}$, and any functions $\tilde{m}_{\Delta}$.
\item If Assumption \ref{asm:SC} holds, then
\begin{align*}
{\Greekmath 0112} & = \mathbb{E}[{\Greekmath 011E}(S;\tilde{m}_{\Delta},p,w;{\Greekmath 0119}_1)],
\end{align*}
for any function $\tilde{m}_{\Delta}$, where $w$ is the set of weights
satisfying (\ref{eqn:w-id}).
\end{enumerate}
\end{theorem}
Theorem \ref{thm:identification} establishes that the moment function ${\Greekmath 011E} $
is unbiased for the ATT across various scenarios,
providing the key sufficient condition for the consistency of the ATT estimator proposed in the next section.
The unbiasedness of ${\Greekmath 011E} $ is
robust, highlighting two layers of double robustness. The first pertains to
identification. The unbiasedness of the moment function$\mathbb{\ }{\Greekmath 011E} $ is
robust to the underlying identification assumption, meaning that $\mathbb{E}
\left( {\Greekmath 011E} \right) $ identifies the ATT as long as either the parallel
trends condition or the synthetic control condition holds. This flexibility
allows researchers to use a single estimator based on ${\Greekmath 011E} $, relieving
them of the burden of choosing between the two methods.
The second, and perhaps more common, notion of double robustness relates to
the specification of nuisance parameters. When the parallel trends condition
holds, the moment function$\mathbb{\ }{\Greekmath 011E} $ remains unbiased as long as
either the outcome model $m_{\Delta }$ or the propensity score model $p$ is
correctly specified. Misspecification of one model will not render the
moment function ${\Greekmath 011E} $ biased as long as the other model is correctly
specified. The double robustness between $p$ and $m$ arises because ${\Greekmath 011E} $
closely resembles the doubly robust DiD moment function in the classical
setting \citep{sant2020doubly}. Moreover, the moment function is unbiased
even if the weights $w$ are entirely misspecified. The weights are
irrelevant for identification since any control group can be used for
comparison on its own, as can any combination of control groups. On the
other hand, under the synthetic control condition, both the propensity
scores $p$ and the weights $w$ must be correctly specified, while the
outcome model $m_{\Delta }$ can be misspecified.
For completeness, we note that an alternative formulation of the moment
function is possible:
\begin{align*}
& {\Greekmath 011E} ^{\#}(S;\{m_{g,\Delta }:2\leq g\leq N_{G}+1\},p,w;{\Greekmath 0119} _{1}) \\
=& \frac{\mathcal{G}_{1}}{{\Greekmath 0119} _{1}}\left( \Delta
Y-\sum_{g=2}^{N_{G}+1}w_{g}(X)m_{g,\Delta }(X)\right) -\frac{1}{{\Greekmath 0119} _{1}}
\sum_{g=2}^{N_{G}+1}w_{g}(X)\frac{p_{1}(X)}{p_{g}(X)}\mathcal{G}_{g}\left(
\Delta Y-m_{g,\Delta }(X)\right) \\
=& \frac{1}{{\Greekmath 0119} _{1}}\sum_{g=2}^{N_{G}+1}w_{g}(X)\left( \mathcal{G}_{1}-
\frac{p_{1}(X)}{p_{g}(X)}\mathcal{G}_{g}\right) (\Delta Y-m_{g,\Delta }(X)),
\end{align*}
where $m_{g,\Delta }(x)\equiv \mathbb{E}[\Delta Y|G=g,X]$. This formulation
offers double robustness between the outcome models $\bigl(m_{g,\Delta
},\,g=2,\ldots,N_{G}+1\bigr)$ and the propensity score model $p$, and it
does so under both the parallel trends and synthetic control assumptions.
However, it requires correct specification of the weights in the synthetic
control setting and therefore does not exhibit Neyman orthogonality with
respect to all nuisance parameters. Moreover, this approach is relatively
non-parsimonious, as it adds another set of nuisance parameters $\bigl(
m_{g,\Delta },\,g=2,\ldots,N_{G}+1\bigr)$, which scales with the number of
groups, on top of the two existing sets of functions $p$ and $w$. Since our
paper focuses on semiparametric estimation of the ATT through nonparametric
first-step estimation guided by identification assumptions (rather than
assuming parametric functional forms), we favor a more parsimonious approach
based on ${\Greekmath 011E} $.
\subsection{Comparison with existing methods in the case without covariates} \label{sec:no-covariate}
As previously mentioned, our framework naturally incorporates conditioning covariates, in contrast to existing causal panel methods. In the absence of covariates, our approach specializes to a formulation directly comparable to leading alternatives, and we provide that comparison in this subsection.
When there is no covariate information, we can simplify the nuisance parameters into constants: $m_{\Delta }(X)\equiv m_{\Delta}$, $p_{1}\left(
X\right) /p_{g}(X)=p_{1}/p_{g}$, and $w_{g}(X)\equiv w_{g}$. For simplicity,
we assume that the number of individuals in group $g$ is $np_{g}.$ Noting
that, in the absence of covariate information, ${\Greekmath 0119} _{1}=p_{1},$ we have
\begin{eqnarray*}
{\Greekmath 011E} (S_{i};m_{\Delta },p,w;{\Greekmath 0119} _{1}) &\equiv &\frac{\mathcal{G}_{1i}}{{\Greekmath 0119}
_{1}}(\Delta Y_{i}-m_{\Delta })-\frac{1}{{\Greekmath 0119} _{1}}\sum_{g=2}^{N_{G}+1}w_{g}
\frac{p_{1}}{p_{g}}\mathcal{G}_{gi}(\Delta Y_{i}-m_{\Delta }) \\
&=&\frac{\mathcal{G}_{1i}}{p_{1}}(\Delta Y_{i}-m_{\Delta
})-\sum_{g=2}^{N_{G}+1}w_{g}\frac{\mathcal{G}_{gi}}{p_{g}}(\Delta
Y_{i}-m_{\Delta }).
\end{eqnarray*}
The simple average estimator based on the above ${\Greekmath 011E} $ is then given by
\begin{eqnarray}
&&\frac{1}{n}\sum_{i=1}^{n}{\Greekmath 011E} (S_{i};m_{\Delta },p,w;{\Greekmath 0119} _{1}) \notag \\
&=&\frac{1}{np_{1}}\sum_{i=1}^{n}\mathcal{G}_{1i}(\Delta Y_{i}-m_{\Delta
})-\sum_{g=2}^{N_{G}+1}w_{g}\frac{1}{np_{g}}\sum_{i=1}^{n}\mathcal{G}
_{gi}(\Delta Y_{i}-m_{\Delta }) \notag \\
&=&(\Delta \bar{Y}_{1}-m_{\Delta })-\sum_{g=2}^{N_{G}+1}w_{g}(\Delta \bar{Y}
_{g}-m_{\Delta })=\Delta \bar{Y}_{1}-\sum_{g=2}^{N_{G}+1}w_{g}\Delta \bar{Y}
_{g} \notag \\
&=&\bar{Y}_{1,\mathcal{T}}-\bar{Y}_{1,\mathcal{T}-1}-
\sum_{g=2}^{N_{G}+1}w_{g}(\bar{Y}_{g,\mathcal{T}}-\bar{Y}_{g,\mathcal{T}-1}),
\label{eqn:degenerate-formula}
\end{eqnarray}
where $\bar{Y}_{g,t}$ is the average of the outcomes for individuals in
group $g$ at time period $t.$
Building on \cite{shen2023same}'s formulation (specifically, their equation
(6)) of the synthetic DiD estimator studied by \cite
{arkhangelsky2021synthetic}, the synthetic DiD (SDiD) estimator of the ATT
based on the group-level data $\{ \bar{Y}_{g,t}:g=1,\ldots,N_{G}+1, t=1,\ldots,\mathcal{T}\} $ is
\begin{equation}
\bar{Y}_{1,\mathcal{T}}-\sum_{g=2}^{N_{G}+1}w_{g}\bar{Y}_{g,\mathcal{T}
}-\sum_{t=1}^{\mathcal{T}-1}{\Greekmath 010B} _{t}\bar{Y}_{1,t}+\sum_{g=2}^{N_{G}+1}
\sum_{t=1}^{\mathcal{T}-1}w_{g}{\Greekmath 010B} _{t}\bar{Y}_{g,t}, \label{eqn:syn_did}
\end{equation}
where ${\Greekmath 010B} _{t}$ represents the coefficient from the horizontal
regression of $\bar{Y}_{\mathcal{T}}$ on its lagged values. With a suitably
chosen time-series prediction for the outcomes at $t=\mathcal{T}$, the
augmented synthetic control (ASC) estimator of \cite{ben2021augmented} also
takes the above form. Note that if we set the temporal weights ${\Greekmath 010B} _{t}$
to be $\mathbf{1}\{t=\mathcal{T}-1\}$, then the estimator in (\ref
{eqn:syn_did}) becomes our estimator in (\ref{eqn:degenerate-formula}).
Theorem 1 of \cite{shen2023same} shows that horizontal and vertical OLS
regressions with the minimum $\ell _{2}$-norm solution, when necessary,
produce a numerically identical estimate of the ATT, which is also
numerically identical to the corresponding SDiD and ASC estimates.\footnote{
The numerical equivalence of the four methods also holds if the weights $
\left\{ {\Greekmath 010B} _{t}\right\} $ and $\left\{ w_{g}\right\} $ are estimated
from a ridge regression with an $\ell _{2}$ penalty. Their difference lies
in how the weights are estimated.} In contrast, our estimator integrates a
first-difference comparison with vertical regression, distinguishing it from
existing approaches, including horizontal and vertical regressions, SDiD,
and ASC. While we adopt the \textquotedblleft similar units behave
similarly\textquotedblright\ principle from vertical regression, we depart
from horizontal regression's reliance on the entire history to guide the
future. Instead, we employ the notion that \textquotedblleft comparable
trends evolve similarly,\textquotedblright\ focusing on trends derived from
only two time periods: the pre-treatment and post-treatment periods. In
addition, it is useful to reiterate that we accommodate covariate
information at the individual level, while existing SC and SDiD methods typically operate on aggregate-level data and can only incorporate aggregate-level covariates.
\section{Estimation and inference}
\label{sec:asymptotics}
The identification result in Theorem \ref{thm:identification} establishes
that estimators based on ${\Greekmath 011E}$ are consistent for ${\Greekmath 0112}$ as long as
either Assumption \ref{asm:PT} or Assumption \ref{asm:SC} is satisfied.
However, the distribution of the estimator varies depending on the
underlying identification assumption. In this section, we derive the
asymptotic distributions under each assumption and present a unified
multiplier bootstrap method for inference.
\subsection{Semiparametric estimation}
Suppose that we have decided on a nonparametric method for estimating each
of $m_{\Delta }$, $p$, and $w$ (we will specify each method in detail
later). We implement the following cross-fitting procedure: Equally divide
the data along the cross-sectional dimension into $L$ folds with the size of
each fold being $n/L$. For notational simplicity, we assume that $n/L$ is an
integer. For $\ell =1,\ldots ,L$, let $I_{\ell }$ denote the index set of
the cross-sectional units in the $\ell $th fold and $I_{\ell
}^{c}=\bigcup_{\ell ^{\prime }\neq \ell }I_{\ell ^{\prime }}$ the index set
of the cross-sectional units not in the $\ell $th fold. For an observation $
X_{i}$ with index $i\in I_{\ell }$, we use the subsample with indices in $
I_{\ell }^{c}$ to construct the nonparametric estimates $\hat{m}_{\Delta
}^{\ell }(X_{i})$, $\hat{p}^{\ell }(X_{i})$, and $\hat{w}^{\ell }(X_{i})$,
where the superscript $\ell $ signifies the fact that each of the three
nonparametric estimators is constructed using data in $I_{\ell }^{c}$. The
semiparametric estimator of ${\Greekmath 0112} $ is constructed as
\begin{equation}
\hat{{\Greekmath 0112}}=\frac{1}{n}\sum_{\ell =1}^{L}\sum_{i\in I_{\ell }}{\Greekmath 011E} (S_{i};
\hat{m}_{\Delta }^{\ell },\hat{p}^{\ell },\hat{w}^{\ell };\hat{{\Greekmath 0119}}_{1}),
\label{eqn:theta-hat-def}
\end{equation}
where $\hat{{\Greekmath 0119}}_{1}=\sum_{i=1}^{n}\mathcal{G}_{1i}/n$ is the sample average
estimator for ${\Greekmath 0119} _{1}$.
\paragraph{Asymptotic theory under parallel trends}
In the absence of Assumption \ref{asm:SC}, when only Assumption \ref{asm:PT}
holds, the synthetic weight $w$ may not be well-defined, meaning that there
may not exist any $w$ such that (\ref{eqn:w-id}) is satisfied. However, for
deriving the asymptotic distribution of $\hat{{\Greekmath 0112}}$, it is necessary for
the random quantity $\hat{w}$ to converge to a probability limit ${\Greekmath 0121}
\equiv \operatorname*{plim} \hat{w}$. This limit ${\Greekmath 0121}$ can be interpreted as a
pseudo-true set of weights that minimizes the discrepancy between the
left-hand and right-hand sides of (\ref{eqn:w-id}).
The following theorem establishes the asymptotic distribution of the
estimator under the parallel trends condition. Throughout this paper,
asymptotics are considered with a fixed number of groups $N_{G}$, time
periods $\mathcal{T}$, and cross-fitting folds $L$, while the
cross-sectional sample size $n$ grows to infinity. To\ simplify the
presentation, we let $w_{1}\left( \cdot \right) $ and $\hat{w}_{1}\left(
\cdot \right) $ be the constant function $\mathbf{1}(\cdot )$ with $\mathbf{1
}(x)=1$ for all $x\in \mathcal{X}$, a convention that will be used
throughout the rest of the paper.
\begin{theorem}
\label{thm:asymp-dist-1} Let Assumptions \ref{asm:NA}, \ref{asm:panel}, \ref
{asm:overlap}, \ref{asm:PT}, and the following conditions hold:
\begin{enumerate}
\item For each $g\geq 1$, $\mathbb{E}[(\Delta Y-m_{\Delta }(X))^{2}|X,G=g]$,
$1/p_{g}$, and ${\Greekmath 0121} _{g}$ are bounded functions in $X$.\footnote{
Assuming that $1/p_{g}$ is bounded for all $g$ implies that the minimum
group size approaches infinity almost surely.}
\item The first-stage estimators satisfy that (1) the estimated weights sum
to one and are bounded in probability, i.e., $\sum_{g\geq 2}\hat{w}
_{g}\left( x\right) =1$ for all $x\in \mathcal{X}$ and $\lVert \hat{w}
_{g}\rVert _{\infty }=O_{p}(1)$ for all $g$, (2) the estimators are $L_{2}$
-consistent, i.e.,
\begin{align*}
\left\Vert \hat{m}_{\Delta }-m_{\Delta }\right\Vert _{L_{2}(F_{X})}&
=o_{p}(1), \\
\left\Vert \hat{p}_{1}/\hat{p}_{g}-p_{1}/p_{g}\right\Vert _{L_{2}(F_{X})}&
=o_{p}(1),\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ }g\geq 2, \\
\left\Vert \hat{w}_{g}-{\Greekmath 0121} _{g}\right\Vert _{L_{2}(F_{X})}& =o_{p}(1),
\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ }g\geq 2,
\end{align*}
and (3) their rates satisfy that
\begin{equation*}
\left\Vert \hat{p}_{1}/\hat{p}_{g}-p_{1}/p_{g}\right\Vert
_{L_{2}(F_{X})}\left\Vert \hat{m}_{\Delta }-m_{\Delta }\right\Vert
_{L_{2}(F_{X})}=o_{p}(n^{-1/2}),\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ }g\geq 2.
\end{equation*}
\end{enumerate}
Then $\sqrt{n}(\hat{{\Greekmath 0112}}-{\Greekmath 0112} )\overset{d}{\rightarrow }N(0,V_{\mathrm{
PT}})$, where the asymptotic variance is
\begin{align}
V_{\mathrm{PT}}& \equiv \frac{1}{{\Greekmath 0119} _{1}^{2}}\Bigg(\mathbb{E}\Bigg[\mathcal{
G}_{1}(\Delta Y-m_{1,\Delta }(X))^{2}+\mathcal{G}_{1}(m_{1,\Delta
}(X)-m_{\Delta }(X)-{\Greekmath 0112} )^{2} \notag \label{eqn:V1} \\
& \quad +\sum_{g=2}^{N_{G}+1}{\Greekmath 0121} _{g}(X)^{2}\mathcal{G}_{g}\frac{
p_{1}(X)^{2}}{p_{g}(X)^{2}}(\Delta Y-m_{\Delta }(X))^{2}\Bigg]\Bigg),
\end{align}
with $m_{1,\Delta }\equiv \mathbb{E}[\Delta Y|X,G=1]$.
\end{theorem}
The asymptotic variance $V_{\mathrm{PT}}$ in (\ref{eqn:V1}) can be expressed
as the variance of the weighted average of the efficient influence functions
for the ATT in a canonical $2\times 2$ DiD design, as characterized in
Proposition 1 of \cite{sant2020doubly}. More specifically, in the $2\times 2$
design with a single control group $g$, the efficient influence function for
the ATT is
\begin{align*}
\mathbb{IF}_{g}& \equiv \frac{1}{{\Greekmath 0119} _{1}}\Bigg(\mathcal{G}_{1}(\Delta
Y-m_{1,\Delta }(X))+\mathcal{G}_{1}(m_{1,\Delta }(X)-m_{\Delta }(X)-{\Greekmath 0112} )
\\
& \quad +\mathcal{G}_{g}\frac{p_{1}(X)}{p_{g}(X)}(\Delta Y-m_{\Delta }(X))
\Bigg).
\end{align*}
Equation (\ref{eqn:V1-calculation}) in the proof shows that $V_{\mathrm{PT}}$
is the variance of the weighted average of all such $\mathbb{IF}_{g}$, that
is, $V_{\mathrm{PT}}=\mathbb{E}[(\sum_{g\geq 2}w_{g}(X)\mathbb{IF}_{g})^{2}]$
. For illustration, in a stylized scenario with a single control group ($g=2$
) and a trivial weight $w_{g}\equiv 1$, $V_{\mathrm{PT}}$ reduces to the
efficiency bound derived in \cite{sant2020doubly}. Alternatively, if one
disregards the finer group structure and instead pools all control groups
into a single aggregated control group, the asymptotic variance of the
doubly robust DiD estimator in \cite{sant2020doubly} becomes
\begin{align}
V_{\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{\textrm{pool}}}& =\frac{1}{{\Greekmath 0119} _{1}^{2}}\Bigg(\mathbb{E}\Bigg[
\mathcal{G}_{1}(\Delta Y-m_{1,\Delta }(X))^{2}+\mathcal{G}_{1}(m_{1,\Delta
}(X)-m_{\Delta }(X)-{\Greekmath 0112} )^{2} \notag \label{eqn:drdid-var} \\
& \quad +\sum_{g=2}^{N_{G}+1}\left( \frac{p_{g}(X)}{p_{-1}(X)}\right) ^{2}
\mathcal{G}_{g}\frac{p_{1}(X)^{2}}{p_{g}(X)^{2}}(\Delta Y-m_{\Delta }(X))^{2}
\Bigg]\Bigg),
\end{align}
where $p_{-1}(X)\equiv \mathbb{P}(G\neq 1|X)$ and the weight $
p_{g}(X)/p_{-1}(X)$ reflects the fraction of units in control group $g$
relative to all control units. The variance $V_{\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{\textrm{pool}}}$ in (
\ref{eqn:drdid-var}) uses the weight $p_{g}(X)/p_{-1}(X)$, which is distinct
from the synthetic control weight ${\Greekmath 0121} _{g}\left( X\right) $ in this
paper. In general, there is no dominance relation between $V_{\mathrm{PT}}$
and $V_{\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{\textrm{pool}}}$.
Theorem \ref{thm:asymp-dist-1} aligns with the framework of double/debiased
machine learning (DML) as discussed in \citep{chernozhukov2018DML}. Under
Assumption \ref{asm:PT}, the moment function ${\Greekmath 011E} $ satisfies Neyman
orthogonality with respect to the nuisance parameters. This implies that the
asymptotic distribution of the estimator is the same as if the true values
of the nuisance parameters were used, provided that the product of the
estimation errors for $m_{\Delta }$ and $p$ converges at a rate faster than $
1/\sqrt{n}$. This rate requirement allows for the use of a wide range of
estimators, including machine learning methods or traditional nonparametric
methods, such as kernel-based or sieve-based methods. Notably, there is no
rate requirement for the weight estimator $w$, as long as it converges in
probability to some limit. This is consistent with the result of Theorem \ref
{thm:identification}(i), which shows that the nuisance weights play a
secondary role under Assumption \ref{asm:PT}.
\paragraph{Asymptotic theory under synthetic control}
Deriving the asymptotic distribution of $\hat{{\Greekmath 0112}}$ under Assumption \ref
{asm:SC} is more challenging because ${\Greekmath 011E} $ does not satisfy Neyman
orthogonality with respect to $p$ and $w$. Consequently, when computing the
asymptotic variance, additional adjustment terms are needed to account for
the first-stage estimation error in estimating these nuisance parameters. To
derive the adjustment terms, we need to find the influence function
associated with their estimation. To keep our paper focused, we examine
kernel-based estimators for the nuisance parameters, although other
nonparametric methods, such as the method of sieves, could also be used.
For any function $f$, define the empirical average operator $P_{n}\left[
\cdot \right] $ as $P_{n}[f(S)]\equiv \frac{1}{n}\sum_{i=1}^{n}f(S_{i})$.
Note that the propensity score enters the estimand only through the ratios $
r_{1,g}\equiv p_{1}/p_{g}$. Using the fact that the ratio $r_{1,g}$
minimizes the objective function $\mathbb{E}[{\Greekmath 011A} (r(X),\mathcal{G})]$ for
\begin{equation*}
{\Greekmath 011A} (r(X),\mathcal{G})\equiv r(X)^{2}\mathcal{G}_{g}-2r(X)\mathcal{G}_{1},
\end{equation*}
we construct the following local polynomial regression:\footnote{
An alternative approach involves estimating the propensity scores using local polynomial regression and combining them into ratio estimates. Under suitable regularity conditions, this estimator has the same Bahadur representation as in Theorem \ref{thm:asymp-dist-2}(ii).}
\begin{equation}
\hat{r}_{1,g}(x)={\Greekmath 0113} _{1}^{\prime }\operatorname*{argmin}_{({\Greekmath 010C} _{0},\ldots ,{\Greekmath 010C} _{
\bar{s}})}P_{n}\left[ K\left( \frac{X-x}{h}\right) {\Greekmath 011A} \left( \sum_{s=0}^{
\bar{s}}{\Greekmath 010C} _{s}(X-x)^{s},\mathcal{G}\right) \right] ,
\label{eqn:hat_r-def}
\end{equation}
where $h$ is the bandwidth, $K$ is the kernel function, ${\Greekmath 0113} _{1}\equiv
(1,0,\ldots ,0)^{\prime }$ is a vector of length $\bar{s}+1$ with $1$ in the
first position and $0$ elsewhere, and $\bar{s}$ is the order of the local
polynomial.\footnote{
Here, we focus on the case with a scalar $X.$ The general case with a vector
$X$ requires only notational changes.} To simplify the presentation, we
define $r_{1,g}(\cdot )$ and $\hat{r}_{1,g}(\cdot )$ to be the constant
function $\mathbf{1}(\cdot )$ when $g=1.$
Second, the weights $w$ are determined by solving the system of
identification equations involving the outcome functions $m_{g,t}(x)\equiv
\mathbb{E}[Y_{t}|G=g,X=x]$. Let $\bm{w}_{0}\equiv (w_{2},\dots
,w_{N_{G}})^{\prime }$ denote the vector of weights excluding the entry for
the last group. Since the weights sum to one, the weight for the last group
can be expressed as $w_{N_{G}+1}=1-\mathbf{1}_{N_{G}-1}^{\prime }\bm{w}_{0}$
, where $\mathbf{1}_{N_{G}-1}$ is a vector of ones with length $N_{G}-1$. By
Assumption \ref{asm:SC}, the weights can be identified via the equation:
\begin{equation*}
\begin{pmatrix}
m_{2,1} & \cdots & m_{N_{G}+1,1} \\
\vdots & & \vdots \\
m_{2,\mathcal{T}-1} & \cdots & m_{N_{G}+1,\mathcal{T}-1}
\end{pmatrix}
\begin{pmatrix}
\bm{w}_{0}^{\prime } \\
1-\mathbf{1}_{N_{G}-1}^{\prime }\bm{w}_{0}
\end{pmatrix}
=
\begin{pmatrix}
m_{1,1} \\
\vdots \\
m_{1,\mathcal{T}-1}.
\end{pmatrix}
\end{equation*}
After rearranging terms, we obtain that $M\bm{w}_{0}=m_{1}$, where
\begin{equation*}
M\equiv
\begin{pmatrix}
m_{-1,1}^{\prime } \\
m_{-1,2}^{\prime } \\
\vdots \\
m_{-1,\mathcal{T}-1}^{\prime }
\end{pmatrix}
,m_{1}\equiv
\begin{pmatrix}
m_{1,1}-m_{N_{G}+1,1} \\
\vdots \\
m_{1,\mathcal{T}-1}-m_{N_{G}+1,\mathcal{T}-1}
\end{pmatrix}
,
\end{equation*}
with $m_{-1,t}\equiv (m_{2,t}-m_{N_{G}+1,t},\ldots
,m_{N_{G},t}-m_{N_{G}+1,t})^{\prime }$. Provided that $\mathcal{T}\geq N_{G}$
and $M^{\prime }M$ is invertible, we can solve for $\bm{w}_{0}$ as
\begin{equation}
\bm{w}_{0}=\operatorname*{argmin}_{u}(Mu-m_{1})^{\prime }(Mu-m_{1})=(M^{\prime
}M)^{-1}M^{\prime }m_{1}. \label{eqn:hat_w-def}
\end{equation}
This is similar to solving an ordinary least squares problem. Let $\hat{M}$
and $\hat{m}_{1}$ denote the respective estimators of $M$ and $m_{1}$,
obtained by replacing each $m_{g,t}$ with the corresponding $\hat{m}_{g,t}$.
The weight estimator is constructed as $\hat{\bm{w}}_{0}\equiv (\hat{M}
^{\prime }\hat{M})^{-1}\hat{M}^{\prime }\hat{m}_{1}$. Thus, the estimation
of $w$ and $m_{\Delta }$ reduces to estimating $m_{g,t}$, which can be
accomplished using another local polynomial regression:
\begin{equation}
\hat{m}_{g,t}(x)={\Greekmath 0113} _{1}^{\prime }\operatorname*{argmin}_{({\Greekmath 010C} _{0},\ldots ,{\Greekmath 010C} _{
\bar{s}})}P_{n}\left[ K\left( \frac{X-x}{h}\right) \left( Y_{t}-\sum_{s=0}^{
\bar{s}}{\Greekmath 010C} _{s}(X-x)^{s}\right) ^{2}\mathcal{G}_{g}\right] .
\label{eqn:hat_m_gt-def}
\end{equation}
\begin{theorem}
\label{thm:asymp-dist-2} Let Assumptions \ref{asm:NA}, \ref{asm:panel}, \ref
{asm:overlap}, \ref{asm:SC}, conditions (i) and (ii) of Theorem \ref
{thm:asymp-dist-1}, and the following conditions hold:
\begin{enumerate}
\item The convergence rates of the nuisance estimators satisfy
\begin{equation*}
\left\Vert \hat{m}_{g,t}-m_{g,t}\right\Vert _{L_{2}(F_{X})},\left\Vert \hat{r
}_{1,g}-r_{1,g}\right\Vert _{L_{2}(F_{X})},\left\Vert \hat{w}
_{g}-w_{g}\right\Vert _{L_{2}(F_{X})}=o_{p}(n^{-1/4}),\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ for all }g.
\end{equation*}
\item Let $h=o(n^{-1/4})$ be an undersmoothing bandwidth, and let $K$ be a
symmetric probability density function satisfying $\int_{-\infty }^{\infty
}u^{2}K(u)du<\infty $. The nuisance estimators admit the following uniform
Bahadur representations: for all $g$,
\begin{align*}
\hat{r}_{1,g}(x)-r_{1,g}(x)& =P_{n}\left[ \frac{K_{h}\left( X-x\right) }{
f_{X}(x)}\frac{\mathcal{G}_{1}-r_{1,g}(X)\mathcal{G}_{g}}{p_{g}(X)}\right]
+o_{p}(n^{-1/2}), \\
\hat{m}_{g,t}(x)-m_{g,t}(x)& =P_{n}\left[ \frac{K_{h}\left( X-x\right) }{
f_{X}(x)}\frac{\mathcal{G}_{g}(Y_{t}-m_{g,t}(X))}{p_{g}(X)}\right]
+o_{p}(n^{-1/2}),1\leq t\leq \mathcal{T}-1,
\end{align*}
where $f_{X}$ is the marginal density function of $X$, $K_{h}(\cdot )\equiv
K(\cdot /h)/h$, and the $o_{p}$-terms hold uniformly over $x$ $\in \mathcal{X
}$, which is assumed to be compact.
\item The functions $w_{g}$, $m_{g,t}$, and $p_{g}$, $1\leq g\leq
N_{G}+1,1\leq t\leq \mathcal{T},$ are twice continuously differentiable.
There exists a constant $c>0$ such that the smallest eigenvalue of $
M(x)^{\prime }M(x)$ is larger than $c$ for all $x\in \mathcal{X}$.
\end{enumerate}
Let ${\Greekmath 0120} _{\mathrm{SC}}\left( S\right) $ be the influence function defined
in (\ref{eqn:psi_SC}) in the proof, and assume that $V_{\mathrm{SC}}=\mathbb{
E}\left[ {\Greekmath 0120} _{\mathrm{SC}}\left( S\right) ^{2}\right] $\ is finite. Then $
\sqrt{n}(\hat{{\Greekmath 0112}}-{\Greekmath 0112} )\overset{d}{\rightarrow }N(0,V_{\mathrm{SC}})$.
\end{theorem}
Since ${\Greekmath 011E} $ is not Neyman orthogonal under Assumption \ref{asm:SC}, the
required rate of convergence for each nuisance function estimator, specified
in the first condition of Theorem \ref{thm:asymp-dist-2}, has been
strengthened to $o_{p}\left( n^{-1/4}\right) $ compared to Theorem \ref
{thm:asymp-dist-1}. This rate is typical for first-step nonparametric
estimators \citep{newey1994asymptotic,chen2003estimation}. The second
condition of Theorem \ref{thm:asymp-dist-2} specifies the asymptotic linear
(Bahadur) representations of local polynomial estimators. These
representations are well-established in the literature, with primitive
conditions provided in, for example, \cite{kong2010uniform}.
The asymptotic properties of our estimator fall within the broader framework
of semiparametric two-step estimation theory
\citep[e.g.,][]{chen2003estimation}. Our main contribution in Theorem \ref
{thm:asymp-dist-2} is to derive the adjustment to the asymptotic variance to
account for the nuisance estimators, which corresponds to Condition (2.6) in
\cite{chen2003estimation}. This derivation is particularly challenging in
our setting due to the sophisticated way in which the nuisance conditional
mean functions $m_{g,t}\left( \cdot \right) $ enter the estimating equation
through the synthetic control weights.
While Theorem \ref{thm:asymp-dist-2} derives the asymptotic distribution of $
\hat{{\Greekmath 0112}}$ under a local polynomial specification for the nuisance
estimators, our result is expected to extend more broadly. The same
asymptotic distribution should hold when alternative nonparametric
estimation methods, such as sieve estimators, are employed. This is because,
according to \cite{newey1994asymptotic}, the adjustment term in the
asymptotic variance should be the same regardless of the estimator used,
though the technical details may differ.
The nonparametric nuisance estimators discussed thus far pertain to the case
of continuous covariates. When discrete covariates are present, a natural
approach is to partition the data according to their levels and perform the
estimation procedure separately within each partition. The theoretical
framework remains valid for each subgroup defined by the discrete
covariates. This approach will be implemented in our empirical analysis in
Section \ref{sec:empirical}.
\subsection{Bootstrap inference}
To conduct inference, we propose a multiplier bootstrap method to
approximate the asymptotic distribution of $\hat{{\Greekmath 0112}}$. While analytical
standard error estimators can be derived in principle, they involve
complicated expressions, particularly under the synthetic control condition
(cf. the asymptotic variance $V_{\mathrm{SC}}$). More importantly, these variance
formulas depend on the identification assumption, which is unknown in
practice. The bootstrap method circumvents the complex variance formulas and
provides a unified approach to inference, as the bootstrap asymptotic
distribution converges to the true asymptotic distribution of the estimator,
regardless of the identification assumption.
Let $\mathcal{W}_{n}\equiv (W_{1},\ldots,W_{n})$ denote the bootstrap
weights. For any function $f$, define the bootstrap operator $P_{n}^{\ast
}[\cdot]$ as $P_{n}^{\ast }[f(S)]\equiv \frac{1}{n}
\sum_{i=1}^{n}W_{i}f(S_{i})$, which represents an empirical operator
constructed using the bootstrap weights, and should not be confused with any
empirical measure. Let $\hat{{\Greekmath 0119}}_{1}^{\ast }\equiv P_{n}^{\ast }[\mathcal{G}
_{1}]$ denote the bootstrap estimator for ${\Greekmath 0119}_{1}$.
To construct the bootstrap nuisance estimators $\hat{r}_{1,g}^{\ast }$ and $
\hat{m}_{g,t}^{\ast }$, we follow the same procedure as the local polynomial
estimators in equations (\ref{eqn:hat_r-def}) to (\ref{eqn:hat_m_gt-def}),
but replace the empirical operator $P_{n}$ with the bootstrap operator $
P_{n}^{\ast }$. The bootstrap synthetic weights estimator $\hat{w}_{g}^{\ast
}$ is constructed using (\ref{eqn:hat_w-def}), replacing $m_{g,t}$ with $
\hat{m}_{g,t}^{\ast }$. In the multiplier bootstrap process, we use the same
sample splitting procedure as before. Specifically, for an observation $
X_{i} $ with index $i\in I_{\ell }$, we use the subsamples $S_{i}$ and
weights $W_{i}$ with indices in $I_{\ell }^{c}$ to compute the nonparametric
estimates $\hat{m}_{g,t}^{\ast ,\ell }(X_{i})$, $\hat{r}_{1,g}^{\ast ,\ell
}(X_{i})$, and $\hat{w}^{\ast ,\ell }(X_{i})$. The bootstrap estimator for $
{\Greekmath 0112} $ is
\begin{equation}
\hat{{\Greekmath 0112}}^{\ast }\equiv \frac{1}{n}\sum_{\ell =1}^{L}\sum_{i\in I_{\ell
}}W_{i}{\Greekmath 011E} (S_{i};\hat{m}_{\Delta }^{\ast ,\ell },\hat{r}^{\ast ,\ell },
\hat{w}^{\ast ,\ell };\hat{{\Greekmath 0119}}_{1}^{\ast }). \label{eqn:theta-hat-star-def}
\end{equation}
With a slight abuse of notation, we use, in the above, the ratios $\hat{r}^{\ast ,\ell }\equiv
(\hat{r}_{1,g}^{\ast ,\ell },g=2,\ldots ,N_{G}+1)$ in place of the
raw propensity scores in ${\Greekmath 011E}$.
The bootstrap weights are required to satisfy the following mild conditions.
\begin{comment}
\begin{namedassumption}{BW}[Bootstrap Weights]
\label{asm:bootstrap-weights}
The bootstrap weights $\{W_i\}_{i=1}^n$ are iid across $i=1,\ldots, n$ and independent from the data $\mathcal{S}_n = (S_i, 1 \leq i \leq n)$. In addition,
\(\mathbb{E}[W_i] = 1\), \(\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{var}(W_i) = 1\), and there exists some ${\Greekmath 010E}>0$ such that \(\mathbb{E}[|W_i|^{2+{\Greekmath 010E}}] < \infty\).
\end{namedassumption}
\end{comment}
This multiplier bootstrap approach differs from the nonparametric bootstrap,
where the bootstrap weights $(W_{1},\ldots,W_{n})$ follow a multinomial
distribution. Assumption \ref{asm:bootstrap-weights} excludes the
nonparametric bootstrap because its weights are not independent.\ This
dependence introduces additional technical challenges in establishing
bootstrap consistency. Consequently, we consider only the multiplier
bootstrap in this paper. Nonetheless, the multiplier bootstrap remains a
practical, widely used, and theoretically valid inference method, as
demonstrated by our next theorem.
Before presenting the theorem, we introduce the following convention: for
any random variable $\Delta _{n}$ that is a function of both $\mathcal{W}
_{n} $ and $\mathcal{S}_{n}$, we say that $\Delta _{n}=o_{p}(1)$ if it
converges in probability to zero under the joint distribution of $(\mathcal{W
}_{n},\mathcal{S}_{n})$, which follows a product law due to the independence
between $\mathcal{W}_{n}$ and $\mathcal{S}_{n}$. We say that $\Delta
_{n}=O_{p}(1)$ if it is bounded in probability under the joint distribution
of $(\mathcal{W}_{n},\mathcal{S}_{n})$.
\begin{theorem}
\label{thm:bootstrap} Let Assumption \ref{asm:bootstrap-weights} hold.
\begin{enumerate}
\item Let the assumptions of Theorem \ref{thm:asymp-dist-1} hold. We further
assume that (1) the elements of $\hat{w}^{\ast }$ sum to one and are bounded
in probability, i.e., $\sum_{g\geq 2}\hat{w}_{g}^{\ast }(x)=1$ for all $x\in
\mathcal{X}$ and $\lVert \hat{w}_{g}^{\ast }\rVert _{\infty }=O_{p}(1),g\geq
2$, (2) the bootstrap nuisance estimators are $L_{2}$-consistent, i.e.,
\begin{align*}
\left\Vert \hat{m}_{\Delta }^{\ast }-m_{\Delta }\right\Vert _{L_{2}(F_{X})}&
=o_{p}(1), \\
\left\Vert \hat{p}_{1}^{\ast }/\hat{p}_{g}^{\ast }-p_{1}/p_{g}\right\Vert
_{L_{2}(F_{X})}& =o_{p}(1),g\geq 2, \\
\left\Vert \hat{w}_{g}^{\ast }-{\Greekmath 0121} _{g}\right\Vert _{L_{2}(F_{X})}&
=o_{p}(1),g\geq 2,
\end{align*}
and (3) their rates satisfy that
\begin{equation*}
\left\Vert \hat{p}_{1}^{\ast }/\hat{p}_{g}^{\ast }-p_{1}/p_{g}\right\Vert
_{L_{2}(F_{X})}\left\Vert \hat{m}_{\Delta }^{\ast }-m_{\Delta }\right\Vert
_{L_{2}(F_{X})}=o_{p}(n^{-1/2}),g\geq 2.
\end{equation*}
In addition, assume that $\mathbb{E}[|{\Greekmath 011E} (S;m_{\Delta },p,w;{\Greekmath 0119}
_{1})-{\Greekmath 0112} \mathcal{G}_{1}/{\Greekmath 0119} _{1}|^{2+{\Greekmath 010E} }]<\infty $ for some
positive ${\Greekmath 010E} $. Then
\begin{equation}
\sup_{a\in \mathbb{R}}\left\vert \mathbb{P}^{\ast }\left( \sqrt{n}(\hat{
{\Greekmath 0112}}^{\ast }-\hat{{\Greekmath 0112}})\leq a\big|\mathcal{S}_{n}\right) -\mathbb{P}
\left( \sqrt{n}(\hat{{\Greekmath 0112}}-{\Greekmath 0112} )\leq a\right) \right\vert =o_{p}(1),
\label{eqn:bootstrap-consistency}
\end{equation}
where $\mathbb{P}^{\ast }(\cdot |\mathcal{S}_{n})$ is the conditional
probability distribution given the sample $\mathcal{S}_{n}$.
\item Let the assumptions of Theorem \ref{thm:asymp-dist-2} hold. We further
assume that the bootstrap nuisance estimators satisfy the following
conditions: for all $g$,
\begin{equation*}
\left\Vert \hat{m}_{g,t}^{\ast }-m_{g,t}\right\Vert
_{L_{2}(F_{X})},\left\Vert \hat{r}_{1,g}^{\ast }-r_{1,g}\right\Vert
_{L_{2}(F_{X})},\left\Vert \hat{w}_{g}^{\ast }-w_{g}\right\Vert
_{L_{2}(F_{X})}=o_{p}(n^{-1/4}),
\end{equation*}
and
\begin{align*}
\hat{r}_{1,g}^{\ast }(x)-r_{1,g}(x)& =P_{n}^{\ast }\left[ \frac{K_{h}\left(
X-x\right) }{f_{X}(x)}\frac{\mathcal{G}_{1}-r_{1,g}(X)\mathcal{G}_{g}}{
p_{g}(X)}\right] +o_{p}(n^{-1/2}), \\
\hat{m}_{g,t}^{\ast }(x)-m_{g,t}(x)& =P_{n}^{\ast }\left[ \frac{K_{h}\left(
X-x\right) }{f_{X}(x)}\frac{\mathcal{G}_{g}(Y_{t}-m_{g,t}(X))}{p_{g}(X)}
\right] +o_{p}(n^{-1/2}),1\leq t\leq \mathcal{T}-1,
\end{align*}
uniformly over $x\in \mathcal{X}$.\footnote{
We define $\hat{r}_{1,g}^{\ast }\left( \cdot \right) $ and $\hat{w}
_{g}^{\ast }\left( \cdot \right) $ to be the constant function $\boldsymbol{1
}\left( \cdot \right) $ when $g=1.$} In addition, assume that the influence
function ${\Greekmath 0120} _{\mathrm{SC}}\left( S\right) $ defined in (\ref{eqn:psi_SC})
has finite $2+{\Greekmath 010E} $ moment for some positive ${\Greekmath 010E} $. Then (\ref
{eqn:bootstrap-consistency}) holds.
\end{enumerate}
\end{theorem}
In Theorem \ref{thm:bootstrap}, all the $o_{p}\left( \cdot \right) $ and $
O_{p}\left( \cdot \right) $ terms in the conditions are understood to hold
under the joint distribution of the bootstrap weights $\mathcal{W}_{n}$ and
the sample $\mathcal{S}_{n}$. These conditions are straightforward to
verify, as the multiplier bootstrap weights are independent of the sample.
In the bootstrap setting, there is another notion of convergence in
probability, which we use in the proof of Theorem \ref{thm:bootstrap}. We
say that $\Delta _{n}=o_{p}^{\ast }(1)$ if for any ${\Greekmath 010E} >0$, $\mathbb{P}
^{\ast }(|\Delta _{n}|>{\Greekmath 010E} $ $|$ $\mathcal{S}_{n})=o_{p}(1)$. Similarly,
there is another notion of boundedness in probability. We say that $\Delta
_{n}=O_{p}^{\ast }(1)$ if for any ${\Greekmath 010E} >0$, there exists a positive
constant $C_{{\Greekmath 010E} }$ such that $\mathbb{P}(\mathbb{P}^{\ast }(|\Delta
_{n}|>C_{{\Greekmath 010E} }|\mathcal{S}_{n})>{\Greekmath 010E} )\rightarrow 0$. In the proof of
the theorem, we utilize Lemma 3 from \cite{cheng2010bootstrap} to convert $
o_{p}(1)$ and $O_{p}(1)$ terms, respectively, into $o_{p}^{\ast }(1)$ and $
O_{p}^{\ast }(1)$ terms, thereby establishing the result of conditional weak
convergence.
The difference between the bootstrap theory of the estimator under parallel
trends and synthetic control is analogous to that in estimation theory.
Under the parallel trends assumption, the conditions on the bootstrap
estimators of the nuisance functions are relatively weaker (only requiring
rate conditions) due to orthogonality.\footnote{
Notably, these bootstrap estimators of the nuisance functions can coincide
with the original sample estimators in Theorem \ref{thm:asymp-dist-1}, as
they automatically satisfy the rate conditions. This is because the moment
function is insensitive to the nuisance estimators. However, this approach
cannot be implemented in practice, as it fails under the synthetic control
condition.} Under the synthetic control assumption, the bootstrap nuisance
estimators are additionally required to satisfy the Bahadur representation
adjusted by the bootstrap weights.
To construct an ${\Greekmath 010B}$-level bootstrap confidence interval based on Theorem \ref{thm:bootstrap}, we take the $\frac{{\Greekmath 010B}}{2}$ and $(1-\frac{{\Greekmath 010B}}{2})$ quantiles of $\hat{{\Greekmath 0112}}^{\ast }-\hat{{\Greekmath 0112}}$, denoted as $(\hat{{\Greekmath 0112}}^{\ast }-\hat{{\Greekmath 0112}})_{\frac{{\Greekmath 010B}}{2}}$ and $(\hat{{\Greekmath 0112}}^{\ast }-\hat{{\Greekmath 0112}})_{1-\frac{{\Greekmath 010B}}{2}}$, respectively. The confidence interval is then given by
\begin{equation*}
\left[ \hat{{\Greekmath 0112}}-(\hat{{\Greekmath 0112}}^{\ast }-\hat{{\Greekmath 0112}})_{1-\frac{{\Greekmath 010B}}{2}},\;
\hat{{\Greekmath 0112}}-(\hat{{\Greekmath 0112}}^{\ast }-\hat{{\Greekmath 0112}})_{\frac{{\Greekmath 010B}}{2}}\right].
\end{equation*}
Finally, we emphasize that Theorems \ref{thm:asymp-dist-1}, \ref
{thm:asymp-dist-2}, and \ref{thm:bootstrap} do not describe the asymptotics
of two different estimators. Rather, they analyze the same estimator $\hat{
{\Greekmath 0112}}$ and its bootstrap counterpart, as defined in (\ref
{eqn:theta-hat-def}) and (\ref{eqn:theta-hat-star-def}), respectively, under
different identification assumptions. In addition, we impose different
assumptions on the estimators of the nuisance functions. Under the synthetic
control condition, the requirements for the nuisance estimators are more
explicit --- and potentially more restrictive --- compared to the rate
requirements under the parallel trends assumption, as the former does not
satisfy Neyman orthogonality.
\section{Repeated cross-sectional data}
\label{sec:rc}
In practice, the ideal panel data structure, where we have repeated
observations of the same individuals or units across multiple time periods,
is not always available. Researchers commonly utilize repeated
cross-sectional data as an alternative, where observations are collected
across distinct time periods without tracking the same individuals. This
data structure is prevalent in scenarios where longitudinal linkage is
either inherently impractical or ruled out by design. Notable applications
include administrative records of episodic events, such as insurance claims
\citep{meyer1995workers} and car accidents \citep{cohen2003effects},
longitudinal analyses of fixed-age cohorts over extended periods \citep{corak2001death}, and nationally representative surveys, such as the
Current Population Survey \citep{acemoglu2001consequences} and the Canadian
General Social Surveys \citep{finkelstein2002effect}, where different
individuals are sampled in each wave.
In repeated cross-sectional data, we do not observe $\{Y_{t}:t=1, \ldots,
\mathcal{T}\}$ directly. Instead, each data entry includes a time indicator
variable $T\in \{1,\ldots ,\mathcal{T}\}$, which specifies the time period
for that entry. Denote $\mathfrak{T}_{t}\equiv \mathbf{1}\{T=t\}$. The
observed outcome is then given by
\begin{equation*}
Y\equiv \sum_{t=1}^{\mathcal{T}}\mathfrak{T}_{t}Y_{t}.
\end{equation*}
The observed variables in the repeated cross-sectional dataset are $S^{\func{
rc}}\equiv (Y,G,T,X)$. We impose the following conditions on the sampling
scheme.
\begin{comment}
\begin{namedassumption}{RCS}[Repeated Cross-Sectional Sampling] \label{asm:rc}
We observe an iid sample $\{S^{\operatorname{rc}}_i = (Y_i, G_i, T_i, X_i): 1 \leq i \leq n\}$ of $S^{\operatorname{rc}} = (Y, G, T, X)$.
\end{namedassumption}
\end{comment}
\begin{comment}
\begin{namedassumption}{TI}[Time Invariance] \label{asm:time-invariance}
The joint distribution of the outcomes, covariates, and group indicator is independent of the time indicator $T$, i.e.,
\begin{align*}
(Y_1,\ldots,Y_{\mathcal{T}},X,G)\perp \!\!\! \perp T.
\end{align*}
Furthermore, for each period
$t$, ${\Greekmath 0115}_t \equiv \mathbb{P}(T=t)>0.$
\end{namedassumption}
\end{comment}
Under the repeated cross-sectional sample scheme in Assumption \ref{asm:rc},
$T_{i}$ is a categorical variable indicating the time period to which
observation $i$ belongs, and $\mathfrak{T}_{ti}=\mathbf{1}\{T_{i}=t\}$ is a
binary variable indicating whether observation $i$ belongs to time period $t$
.
Assumption \ref{asm:time-invariance} aligns with the time invariance
conditions commonly found in the DiD literature, such as Assumption 3.3 in
\cite{abadie2005semiparametric}. Under this assumption, the joint
distribution of $(Y,G,X,T)$ can be expressed through the following mixture
representation:
\begin{equation*}
\mathbb{P}(Y\leq y,G=g,X\leq x,T=t)={\Greekmath 0115} _{t}\mathbb{P}(Y_{t}\leq
y,G=g,X\leq x).
\end{equation*}
Similar to the repeated cross-sectional settings discussed in \cite
{sant2020doubly} and \cite{callaway2021difference}, Assumption \ref
{asm:time-invariance} excludes compositional changes, that is, changes in
the distribution of $G$ and $X$ over time. Recent work by \cite
{sant2023difference} has explored compositional changes in the DiD setting.
We leave the investigation of such changes within our framework for future
research.
Since each $Y_{t}$ is not directly observable in repeated cross-sections,
the nuisance parameters $m_{g,t}$ and $m_{\Delta }$ need to be redefined. We
introduce some new notations. Let ${\Greekmath 0116} _{g,t}\left( x\right) \equiv \mathbb{E
}[\mathfrak{T}_{t}Y|G=g,X=x]$ and ${\Greekmath 0116} _{G\neq 1,t}\left( x\right) \equiv
\mathbb{E}[\mathfrak{T}_{t}Y|G\neq 1,X=x]$, which are directly identifiable
based on the observation of $S^{rc}$. Then, under Assumption \ref
{asm:time-invariance},
\begin{align*}
{\Greekmath 0116} _{g,t}\left( x\right) & =\mathbb{E}[\mathfrak{T}_{t}Y|G=g,X=x,\mathfrak{T
}_{t}=1]\Pr \left( \mathfrak{T}_{t}=1|G=g,X=x\right) \\
& =\mathbb{E}[Y|G=g,X=x,\mathfrak{T}_{t}=1]\cdot {\Greekmath 0115} _{t}\equiv
m_{g,t}\left( x\right) {\Greekmath 0115} _{t},
\end{align*}
where $m_{g,t}\left( x\right) =\mathbb{E}[Y|G=g,X=x,\mathfrak{T}_{t}=1].$
Similarly, under Assumption \ref{asm:time-invariance},
\begin{equation*}
{\Greekmath 0116} _{G\neq 1,t}\left( x\right) =\mathbb{E}[Y|G\neq 1,X=x,\mathfrak{T}
_{t}=1]\cdot {\Greekmath 0115} _{t}\equiv m_{G\neq 1,t}\left( x\right) {\Greekmath 0115} _{t},
\end{equation*}
where $m_{G\neq 1,t}\left( x\right) =\mathbb{E}[Y|G\neq 1,X=x,\mathfrak{T}
_{t}=1].$ Define
\begin{equation*}
m_{\Delta }\left( x\right) \equiv m_{G\neq 1,\mathcal{T}}\left( x\right)
-m_{G\neq 1,\mathcal{T}-1}\left( x\right) =\frac{{\Greekmath 0116} _{G\neq 1,\mathcal{T}
}\left( x\right) }{{\Greekmath 0115} _{\mathcal{T}}}-\frac{{\Greekmath 0116} _{G\neq 1,\mathcal{T}
-1}\left( x\right) }{{\Greekmath 0115} _{\mathcal{T}-1}}.
\end{equation*}
We will still maintain Assumption \ref{asm:PT} or \ref{asm:SC}. To
facilitate interpretation, we rewrite them in an equivalent form. For
Assumption \ref{asm:PT}, we express it as:
\begin{align*}
& \mathbb{E}\left[ Y_{\mathcal{T}}\left( 0\right) |G=1,X\right] -\mathbb{E}
\left[ Y_{\mathcal{T-}1}\left( 0\right) |G=1,X\right] \\
=\,& \mathbb{E}\left[ Y_{\mathcal{T}}\left( 0\right) |G=g,X\right] -\mathbb{E
}\left[ Y_{\mathcal{T-}1}\left( 0\right) |G=g,X\right]
\end{align*}
or equivalently
\begin{align*}
& \mathbb{E}\left[ Y\left( 0\right) |G=1,X,\mathfrak{T}_{\mathcal{T}}=1
\right] -\mathbb{E}\left[ Y\left( 0\right) |G=1,X,\mathfrak{T}_{\mathcal{T}
-1}=1\right] \\
=\,& \mathbb{E}\left[ Y\left( 0\right) |G=g,X,\mathfrak{T}_{\mathcal{T}}=1
\right] -\mathbb{E}\left[ Y\left( 0\right) |G=g,X,\mathfrak{T}_{\mathcal{T}
-1}=1\right] .
\end{align*}
This says that the change in the conditional mean of the baseline outcome
for the treatment group from period $\mathcal{T}-1$ to period $\mathcal{T}$
is the same as that for any of the control groups. For each of the two time
periods, the conditional mean is taken only over the observable individuals.
For Assumption \ref{asm:SC}, we express it as
\begin{equation*}
\mathbb{E}\left[ Y\left( 0\right) |G=1,X=x,\mathfrak{T}_{t}=1\right]
=\sum_{g=2}^{N_{G}+1}w_{g}\left( x\right) \mathbb{E}\left[ Y\left( 0\right)
|G=g,X=x,\mathfrak{T}_{t}=1\right] ,
\end{equation*}
for $t=1,\ldots ,\mathcal{T}$. So the synthetic control condition is imposed
on the conditional means of the baseline outcome for the observable
individuals in each group. The weight $w_{g}\left( x\right) $ can be
identified from the following equations:
\begin{equation*}
{\Greekmath 0116} _{1,t}\left( x\right) =\sum_{g=2}^{N_{G}+1}w_{g}\left( x\right) {\Greekmath 0116}
_{g,t}\left( x\right) \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ for }t=1,\ldots ,\mathcal{T}-1,
\end{equation*}
where the factor $1/{\Greekmath 0115} _{t}$ on both sides cancels out. Similar to (\ref
{eqn:hat_w-def}), we can solve for $\bm{w}_{0}$, which is the vector of
weights excluding the last entry, as $\bm{w}_{0}=(M_{\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{rc}}^{\prime }M_{
\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{rc}})^{-1}M_{\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{rc}}^{\prime }m_{1,\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{rc}}$, where
\begin{equation*}
M_{\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{rc}}\equiv
\begin{pmatrix}
{\Greekmath 0116} _{-1,1}^{\prime } \\
{\Greekmath 0116} _{-1,2}^{\prime } \\
\vdots \\
{\Greekmath 0116} _{-1,\mathcal{T}-1}^{\prime }
\end{pmatrix}
,m_{1,\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{rc}}\equiv
\begin{pmatrix}
{\Greekmath 0116} _{1,1}-{\Greekmath 0116} _{N_{G}+1,1} \\
\vdots \\
{\Greekmath 0116} _{1,\mathcal{T}-1}-{\Greekmath 0116} _{N_{G}+1,\mathcal{T}-1}
\end{pmatrix}
,
\end{equation*}
with ${\Greekmath 0116} _{-1,t}\equiv ({\Greekmath 0116} _{2,t}-{\Greekmath 0116} _{N_{G}+1,t},\ldots ,{\Greekmath 0116}
_{N_{G},t}-{\Greekmath 0116} _{N_{G}+1,t})^{\prime }$.
We write the doubly robust moment function for ${\Greekmath 0112} $ in the repeated
cross-sectional setting as
\begin{align*}
& {\Greekmath 011E}_{\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{rc}}(S^{\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{rc}};{\Greekmath 0116} _{G\neq 1,\mathcal{T}},{\Greekmath 0116} _{G\neq 1,
\mathcal{T}-1},p,w;{\Greekmath 0119} _{1},{\Greekmath 0115} _{\mathcal{T}},{\Greekmath 0115} _{\mathcal{T}-1})
\\
=\,& \frac{1}{{\Greekmath 0119} _{1}}\left( \mathcal{G}_{1}-\sum_{g=2}^{N_{G}+1}\mathcal{G}
_{g}w_{g}(X)\frac{p_{1}(X)}{p_{g}(X)}\right) \left( \left( \frac{\mathfrak{T}
_{\mathcal{T}}}{{\Greekmath 0115} _{\mathcal{T}}}-\frac{\mathfrak{T}_{\mathcal{T}-1}}{
{\Greekmath 0115} _{\mathcal{T}-1}}\right) Y-\left( \frac{{\Greekmath 0116} _{G\neq 1,\mathcal{T}}(X)
}{{\Greekmath 0115} _{\mathcal{T}}}-\frac{{\Greekmath 0116} _{G\neq 1,\mathcal{T}-1}(X)}{{\Greekmath 0115} _{
\mathcal{T}-1}}\right) \right) .
\end{align*}
\begin{corollary}
\label{thm:rc-identification} Let Assumptions \ref{asm:NA}, \ref{asm:overlap}
, \ref{asm:rc}, and \ref{asm:time-invariance} hold. \
\begin{enumerate}
\item If Assumption \ref{asm:PT} holds, then
\begin{align*}
{\Greekmath 0112} & =\mathbb{E}[{\Greekmath 011E} _{\mathrm{rc}}(S^{\mathrm{rc}};{\Greekmath 0116} _{G\neq 1,
\mathcal{T}},{\Greekmath 0116} _{G\neq 1,\mathcal{T}-1},\tilde{p},\tilde{w};{\Greekmath 0119}
_{1},{\Greekmath 0115} _{\mathcal{T}},{\Greekmath 0115} _{\mathcal{T}-1})] \\
& =\mathbb{E}[{\Greekmath 011E} _{\mathrm{rc}}(S^{\mathrm{rc}};\tilde{{\Greekmath 0116}}_{G\neq 1,
\mathcal{T}},\tilde{{\Greekmath 0116}}_{G\neq 1,\mathcal{T}-1},p,\tilde{w};{\Greekmath 0119}
_{1},{\Greekmath 0115} _{\mathcal{T}},{\Greekmath 0115} _{\mathcal{T}-1})],
\end{align*}
for any vector of weight functions $\Tilde{w}$ whose elements sum to one,
any non-zero functions $\Tilde{p}$, and any functions $(\tilde{{\Greekmath 0116}}_{G\neq 1,
\mathcal{T}},\tilde{{\Greekmath 0116}}_{G\neq 1,\mathcal{T}-1})$.
\item If Assumption \ref{asm:SC} holds, then
\begin{equation*}
{\Greekmath 0112} =\mathbb{E}[{\Greekmath 011E} _{\mathrm{rc}}(S^{\mathrm{rc}};\tilde{{\Greekmath 0116}}_{G\neq 1,
\mathcal{T}},\tilde{{\Greekmath 0116}}_{G\neq 1,\mathcal{T}-1},p,w;{\Greekmath 0119} _{1},{\Greekmath 0115} _{
\mathcal{T}},{\Greekmath 0115} _{\mathcal{T}-1})],
\end{equation*}
for any functions $(\tilde{{\Greekmath 0116}}_{G\neq 1,\mathcal{T}},\tilde{{\Greekmath 0116}}_{G\neq 1,
\mathcal{T}-1})$, where $w$ is the vector of weight functions satisfying (
\ref{eqn:w-id}).
\end{enumerate}
\end{corollary}
For estimation, we consider the same cross-fitting method as in the panel
setting. The estimator for ${\Greekmath 0112} $ is
\begin{equation*}
\hat{{\Greekmath 0112}}_{\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{rc}}\equiv \frac{1}{n}\sum_{\ell =1}^{L}\sum_{i\in
I_{\ell }}{\Greekmath 011E} _{\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{rc}}(S_{i}^{\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{rc}};\hat{{\Greekmath 0116}}_{G\neq 1,\mathcal{T}
}^{\ell },\hat{{\Greekmath 0116}}_{G\neq 1,\mathcal{T}-1}^{\ell },\hat{p}^{\ell },\hat{w}
^{\ell };\hat{{\Greekmath 0119}}_{1},\hat{{\Greekmath 0115}}_{\mathcal{T}},\hat{{\Greekmath 0115}}_{\mathcal{T}
-1}),
\end{equation*}
where $\hat{{\Greekmath 0116}}_{G\neq 1,\mathcal{T}}^{\ell },\hat{{\Greekmath 0116}}_{G\neq 1,\mathcal{T}
-1}^{\ell },\hat{p}^{\ell },$ and $\hat{w}^{\ell }$ are nuisance estimators
constructed using samples not in the $\ell $th fold, and $\hat{{\Greekmath 0115}}
_{t}\equiv P_{n}\mathfrak{T}_{t}$ is the estimator for ${\Greekmath 0115} _{t}$, for $
t=\mathcal{T}-1,\mathcal{T}$.
\begin{theorem}
\label{thm:rc-PT} Let Assumptions \ref{asm:NA}, \ref{asm:overlap}, \ref
{asm:PT}, \ref{asm:rc}, \ref{asm:time-invariance}, and the following
conditions hold:
\begin{enumerate}
\item For each $g\geq 1$, $\mathbb{E}\Big[\Big(\frac{\mathfrak{T}_{\mathcal{T
}}Y-{\Greekmath 0116} _{G\neq 1,\mathcal{T}}(X)}{{\Greekmath 0115} _{\mathcal{T}}}-\frac{\mathfrak{T}
_{\mathcal{T}-1}Y-{\Greekmath 0116} _{G\neq 1,\mathcal{T}-1}(X)}{{\Greekmath 0115} _{\mathcal{T}-1}}
\Big)^{2}\mid X,G=g\Big]$, $1/p_{g}$, and ${\Greekmath 0121} _{g}$ are bounded
functions in $X$.
\item The first-stage estimators satisfy that (1) the estimated weight
functions sum to one and are bounded in probability, i.e., $\sum_{g\geq 2}
\hat{w}_{g}\left( x\right) =1$ for all $x\in \mathcal{X}$ and $\lVert \hat{w}
_{g}\rVert _{\infty }=O_{p}(1),g\geq 2$, (2) the estimators are $L_{2}$
-consistent, i.e.,
\begin{align*}
\left\Vert \hat{{\Greekmath 0116}}_{G\neq 1,t}-{\Greekmath 0116} _{G\neq 1,t}\right\Vert
_{L_{2}(F_{X})}& =o_{p}(1),t=\mathcal{T}-1,\mathcal{T}, \\
\left\Vert \hat{p}_{1}/\hat{p}_{g}-p_{1}/p_{g}\right\Vert _{L_{2}(F_{X})}&
=o_{p}(1),g\geq 2, \\
\left\Vert \hat{w}_{g}-{\Greekmath 0121} _{g}\right\Vert _{L_{2}(F_{X})}&
=o_{p}(1),g\geq 2,
\end{align*}
and (3) their rates satisfy that
\begin{equation*}
\left\Vert \hat{p}_{1}/\hat{p}_{g}-p_{1}/p_{g}\right\Vert
_{L_{2}(F_{X})}\left\Vert \hat{{\Greekmath 0116}}_{G\neq 1,t}-{\Greekmath 0116} _{G\neq 1,t}\right\Vert
_{L_{2}(F_{X})}=o_{p}(n^{-1/2}),g\geq 2,t=\mathcal{T}-1,\mathcal{T}.
\end{equation*}
\end{enumerate}
Then $\sqrt{n}(\hat{{\Greekmath 0112}}_{\mathrm{rc}}-{\Greekmath 0112} )\overset{d}{\rightarrow }
N(0,V_{\mathrm{PT}}^{\mathrm{rc}})$, where $V_{\mathrm{PT}}^{\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{rc}}=
\mathbb{E}[{\Greekmath 0120} _{\mathrm{PT}}^{\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{rc}}(S^{\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{rc}})^{2}]$ and ${\Greekmath 0120} _{
\mathrm{PT}}^{\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{rc}}(S^{\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{rc}})$ is the influence function given in (
\ref{eqn:IF-theta-rc-PT}) in the proof.
\end{theorem}
\begin{theorem}
\label{thm:rc-SC} Let Assumptions \ref{asm:NA}, \ref{asm:overlap}, \ref
{asm:SC}, \ref{asm:rc}, \ref{asm:time-invariance}, conditions (i) and (ii)
of Theorem \ref{thm:rc-PT}, and the following conditions hold:
\begin{enumerate}
\item The convergence rates of the nuisance estimators satisfy
\begin{equation*}
\left\Vert \hat{{\Greekmath 0116}}_{g,t}-{\Greekmath 0116} _{g,t}\right\Vert _{L_{2}(F_{X})},\left\Vert
\hat{r}_{1,g}-r_{1,g}\right\Vert _{L_{2}(F_{X})},\left\Vert \hat{w}
_{g}-w_{g}\right\Vert _{L_{2}(F_{X})}=o_{p}(n^{-1/4}),\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ for all }g.
\end{equation*}
\item Let $h=o(n^{-1/4})$ be an undersmoothing bandwidth, and let $K$ be a
symmetric probability density function satisfying $\int_{-\infty }^{\infty
}u^{2}K(u)\,du<\infty $. The nuisance estimators admit the following uniform
Bahadur representations: for all $g$,
\begin{align*}
\hat{r}_{1,g}(x)-r_{1,g}(x)& =P_{n}\left[ \frac{K_{h}\left( X-x\right) }{
f_{X}(x)}\frac{\mathcal{G}_{1}-r_{1,g}(X)\mathcal{G}_{g}}{p_{g}(X)}\right]
+o_{p}(n^{-1/2}), \\
\hat{{\Greekmath 0116}}_{g,t}(x)-{\Greekmath 0116} _{g,t}(x)& =P_{n}\left[ \frac{K_{h}\left( X-x\right)
}{f_{X}(x)}\frac{\mathcal{G}_{g}(\mathfrak{T}_{t}Y-{\Greekmath 0116} _{g,t}(X))}{p_{g}(X)}
\right] +o_{p}(n^{-1/2}),1\leq t\leq \mathcal{T}-1,
\end{align*}
where $K_{h}(\cdot )\equiv K(\cdot /h)/h$ and the $o_{p}$-terms hold
uniformly over $x\in $ $\mathcal{X}$, which is assumed to be compact.
\item The functions $w_{g}$, ${\Greekmath 0116} _{g,t}$, and $p_{g}$, for $1\leq g\leq
N_{G}+1,1\leq t\leq \mathcal{T},$ are twice continuously differentiable.
There exists a constant $c>0$ such that the smallest eigenvalue of $M_{
\mathrm{rc}}(x)^{\prime }M_{\mathrm{rc}}(x)$ is larger than $c$ for any $
x\in \mathcal{X}$.
\end{enumerate}
Let ${\Greekmath 0120} _{\mathrm{SC}}^{\mathrm{rc}}(S^{\mathrm{rc}})$ be the influence
function defined in (\ref{eqn:IF-theta-rc-SC}) in the proof, and assume that
$V_{\mathrm{SC}}^{\mathrm{rc}}=\mathbb{E}[{\Greekmath 0120} _{\mathrm{SC}}^{\mathrm{rc}
}(S^{\mathrm{rc}})^{2}]$ is finite. Then $\sqrt{n}(\hat{{\Greekmath 0112}}_{\mathrm{rc}
}-{\Greekmath 0112} )\overset{d}{\rightarrow }N(0,V_{\mathrm{SC}}^{\mathrm{rc}}).$
\end{theorem}
Similar to the panel setting, we construct the bootstrap estimator as
\begin{equation*}
\hat{{\Greekmath 0112}}_{\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{rc}}^{\ast }\equiv \frac{1}{n}\sum_{\ell
=1}^{L}\sum_{i\in I_{\ell }}W_{i}{\Greekmath 011E} _{\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{rc}}(S_{i}^{\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{rc}};\hat{{\Greekmath 0116}
}_{G\neq 1,\mathcal{T}}^{\ast ,\ell },\hat{{\Greekmath 0116}}_{G\neq 1,\mathcal{T}
-1}^{\ast ,\ell },\hat{p}^{\ast ,\ell },\hat{w}^{\ast ,\ell };\hat{{\Greekmath 0119}}
_{1}^{\ast },\hat{{\Greekmath 0115}}_{\mathcal{T}}^{\ast },\hat{{\Greekmath 0115}}_{\mathcal{T}
-1}^{\ast }),
\end{equation*}
where $\hat{{\Greekmath 0115}}_{t}^{\ast }\equiv P_{n}^{\ast }\mathfrak{T}_{t},t=
\mathcal{T}-1,\mathcal{T}$, and the bootstrap nonparametric nuisance
estimators are constructed analogously to the panel setting.
\begin{corollary}
\label{cor:rc-bootstrap} Suppose that the bootstrap weights satisfy
Assumption \ref{asm:bootstrap-weights}.
\begin{enumerate}
\item Let the assumptions of Theorem \ref{thm:rc-PT} hold. We further assume
that the bootstrap nuisance estimators satisfy that (1) the elements of $
\hat{w}^{\ast }$ sum to one and are bounded in probability, i.e., $
\sum_{g\geq 2}\hat{w}_{g}^{\ast }\left( x\right) =1$, $x\in \mathcal{X}$ and
$\lVert \hat{w}_{g}^{\ast }\rVert _{\infty }=O_{p}(1),g\geq 2$, (2) the
bootstrap nuisance estimators are $L_{2}$-consistent, i.e.,
\begin{align*}
\left\Vert \hat{{\Greekmath 0116}}_{G\neq 1,t}^{\ast }-{\Greekmath 0116} _{G\neq 1,t}\right\Vert
_{L_{2}(F_{X})}& =o_{p}(1),t=\mathcal{T}-1,\mathcal{T}, \\
\left\Vert \hat{p}_{1}^{\ast }/\hat{p}_{g}^{\ast }-p_{1}/p_{g}\right\Vert
_{L_{2}(F_{X})}& =o_{p}(1),g\geq 2, \\
\left\Vert \hat{w}_{g}^{\ast }-{\Greekmath 0121} _{g}\right\Vert _{L_{2}(F_{X})}&
=o_{p}(1),g\geq 2,
\end{align*}
and (3) their rates satisfy that
\begin{equation*}
\left\Vert \hat{p}_{1}^{\ast }/\hat{p}_{g}^{\ast }-p_{1}/p_{g}\right\Vert
_{L_{2}(F_{X})}\left\Vert \hat{{\Greekmath 0116}}_{G\neq 1,t}^{\ast }-{\Greekmath 0116} _{G\neq
1,t}\right\Vert _{L_{2}(F_{X})}=o_{p}(n^{-1/2}),g\geq 2,t=\mathcal{T}-1,
\mathcal{T}.
\end{equation*}
In addition, assume that the influence function defined in (\ref
{eqn:IF-theta-rc-PT}) has finite $2+{\Greekmath 010E} $ moment for some positive $
{\Greekmath 010E} $. Then we have
\begin{equation}
\sup_{a\in \mathbb{R}}\left\vert \mathbb{P}^{\ast }\left( \sqrt{n}(\hat{
{\Greekmath 0112}}_{\mathrm{rc}}^{\ast }-\hat{{\Greekmath 0112}}_{\mathrm{rc}})\leq a|\mathcal{S}
_{n}\right) -\mathbb{P}\left( \sqrt{n}(\hat{{\Greekmath 0112}}_{\mathrm{rc}}-{\Greekmath 0112}
)\leq a\right) \right\vert =o_{p}(1). \label{eqn:rc-bootstrap-consistency}
\end{equation}
\item Let the assumptions of Theorem \ref{thm:rc-SC} hold. We further assume
that the bootstrap nuisance estimators satisfy the following conditions: for
all $g$,
\begin{equation*}
\left\Vert \hat{{\Greekmath 0116}}_{g,t}^{\ast }-{\Greekmath 0116} _{g,t}\right\Vert
_{L_{2}(F_{X})},\left\Vert \hat{r}_{1,g}^{\ast }-r_{1,g}\right\Vert
_{L_{2}(F_{X})},\left\Vert \hat{w}_{g}^{\ast }-w_{g}\right\Vert
_{L_{2}(F_{X})}=o_{p}(n^{-1/4}),
\end{equation*}
and
\begin{align*}
\hat{r}_{1,g}^{\ast }(x)-r_{1,g}(x)& =P_{n}^{\ast }\left[ \frac{K_{h}\left(
X-x\right) }{f_{X}(x)}\frac{\mathcal{G}_{1}-r_{1,g}(X)\mathcal{G}_{g}}{
p_{g}(X)}\right] +o_{p}(n^{-1/2}), \\
\hat{{\Greekmath 0116}}_{g,t}^{\ast }(x)-{\Greekmath 0116} _{g,t}(x)& =P_{n}^{\ast }\left[ \frac{
K_{h}\left( X-x\right) }{f_{X}(x)}\frac{\mathcal{G}_{g}(\mathfrak{T}
_{t}Y-{\Greekmath 0116} _{g,t}(X))}{p_{g}(X)}\right] +o_{p}(n^{-1/2}),1\leq t\leq \mathcal{
T}-1,
\end{align*}
uniformly over $x\in \mathcal{X}$. In addition, assume that the influence
function ${\Greekmath 0120} _{\mathrm{SC}}^{\mathrm{rc}}(S^{\mathrm{rc}})$ defined in (
\ref{eqn:IF-theta-rc-SC}) has finite $2+{\Greekmath 010E} $ moment for some positive $
{\Greekmath 010E} $. Then (\ref{eqn:rc-bootstrap-consistency}) holds.
\end{enumerate}
\end{corollary}
\section{Staggered treatment adoption}
\label{sec:staggered}
Staggered design, where different groups adopt the treatment at varying
times, is a common feature of panel datasets. Our methodology can be readily
extended to accommodate such staggered designs. Let $D_{t}$ represent the
treatment indicator in period $t$. In the settings of previous sections, $
D_{t}$ equals one if and only if the unit belongs to the treatment group and
the time period is the final one. Under the staggered design, however, $
D_{t} $ can equal one for any group and in any time period. We focus on the
common case where the treatment status is irreversible, meaning $D_{t}\geq
D_{t-1}$. In other words, once a unit becomes treated, it remains treated
throughout.
The initial treatment time for an individual, $\min \{t: D_t = 1\}$, is
determined by the group to which the individual belongs. Mathematically,
this means that $\min \{t: D_t = 1\}$ is measurable with respect to the
sigma-algebra generated by $G$. In the empirical contexts where states serve
as the aggregate units, this implies that different states will have their
own timing for policy changes. However, once a state implements the change,
the entire state adopts the new policy uniformly. Additionally, multiple
states are permitted to adopt the treatment at the same time. We denote this
mapping from $G$ to $\min \{t: D_t = 1\}$ as ${\Greekmath 010D}(G)$. We adopt the
convention that the minimum element of an empty set is $\infty$. When $
{\Greekmath 010D}(G) = \infty$, the individual remains untreated throughout the entire
time span of the dataset. The eventually treated units are those with $
{\Greekmath 010D}(G) < \infty$.
In general, potential outcomes in a staggered treatment design should be
defined based on all potential treatment trajectories $\left( D_{1},\ldots
,D_{\mathcal{T}}\right) $, and we may denote them as $\tilde{Y}_{t}\left(
d_{1},\ldots ,d_{\mathcal{T}}\right) $. Given the irreversibility of the
treatment, we adopt the simple notation $Y_{t}(s)=\tilde{Y}_{t}\left(
d_{1},\ldots ,d_{\mathcal{T}}\right) $ where $s=\min \{t:d_{t}=1\}.$ That
is, $Y_{t}(s)$ represents the potential outcome at time $t$ had an
individual belonged to a group whose time of treatment adoption was $s$. The observed outcome is $Y_{t}\equiv
Y_{t}(G)=\sum_{g=1}^{N_{G}+1}\mathcal{G}_{g}Y_{t}({\Greekmath 010D}(g))$. Following the
literature, we assume that, prior to treatment, a unit's potential outcomes
are equal to its never-treated potential outcomes, a generalization of the
no-anticipation assumption to the staggered treatment setting.
\begin{comment}
\begin{namedassumption}{NA-stg}[No Anticipation in Staggered Design] \label{asm:NA-staggered}
For $t < {\Greekmath 010D}(g)$, $Y_{t}(g) = Y_t(\infty)$.
\end{namedassumption}
\end{comment}
The causal effects of interest are the group-level ATTs in post-treatment
periods:
\begin{equation*}
\func{ATT}(g,t)\equiv \mathbb{E}[Y_{t}({\Greekmath 010D}(g))-Y_{t}(\infty )|G=g],\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ for }
t\geq {\Greekmath 010D} (g).
\end{equation*}
To analyze how the treatment effect evolves over time, a common approach is
to transform calendar time $t$ into event time $e=t-{\Greekmath 010D} (g)$, which
measures the time elapsed relative to the treatment's initiation for group $
g $. This framework, often referred to as an event study, focuses on the
causal parameter of interest as a weighted average of ATTs:
\begin{align*}
\func{ES}(e)& \equiv \mathbb{E}[\func{ATT}(G,{\Greekmath 010D} (G)+e)|{\Greekmath 010D} (G)+e\in
\lbrack 2,\mathcal{T}]] \\
& =\sum_{g:{\Greekmath 010D} (g)<\infty }\mathbb{P}(G=g|{\Greekmath 010D} (G)+e\in \lbrack 2,
\mathcal{T}])\func{ATT}(g,{\Greekmath 010D} (g)+e),\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ }e\geq 0.
\end{align*}
The ATT parameters are the building blocks of the event study parameters.
Once the identification and estimation of the ATTs are established, the
event study parameters can be derived straightforwardly.
To identify the ATTs, we generalize the parallel trends and synthetic
control assumptions to the staggered design. We need to first specify the
set of control groups for comparison, commonly known as the donor pool in
the literature. For an event time $e\geq 0$, the set of possible donor
groups for the identification of $\func{ATT}(g,{\Greekmath 010D} (g)+e)$ consists of
groups that are untreated up to the period ${\Greekmath 010D} (g)+e$:
\begin{equation*}
\mathcal{D}_{g,e}\equiv \{g^{\prime }:{\Greekmath 010D} (g^{\prime })>{\Greekmath 010D} (g)+e\}.
\end{equation*}
This definition of the donor groups depends on both the treated group $g$
and event time $e$, which could make the method overly complex. To avoid a
time-varying donor group, we follow \cite{ben2022synthetic} and focus on a
fixed set of donor groups $\mathcal{D}_{g,\bar{e}}$ for some sufficiently
large $\bar{e}$ that exceeds all event times of interest. This approach
ensures that the estimated counterfactual outcomes remain stable and are not
artificially affected by variations in the composition of the donor groups
across event time. Nevertheless, our proposed method can easily accommodate
varying donor groups by replacing $\mathcal{D}_{g,\bar{e}}$ with $\mathcal{D}
_{g,e}$ in the procedure. Note that $\mathcal{D}_{g,\bar{e}}$ will always be
non-empty if there are never-treated groups with ${\Greekmath 010D} (g)=\infty $ in the
data.
The following assumptions are maintained for the identification of the
event-study ATTs: $\func{ATT}(g,{\Greekmath 010D} (g)+e)$, with $2\leq {\Greekmath 010D} (g) + e
\leq \mathcal{T} $ and $e\in \{0,1,\ldots ,\bar{e}\}$.
\begin{comment}
\begin{namedassumption}{PT-stg}[Parallel Trends in Staggered Design] \label{asm:PT-staggered}
For any $g$ and $g'$ such that $g' \in \mathcal{D}_{g,\bar{e}}$,
\begin{align*}
\mathbb{E}[Y_t(\infty) - Y_{t-1}(\infty) | G=g,X] = \mathbb{E}[Y_t(\infty) - Y_{t-1}(\infty) | G=g',X], {\Greekmath 010D}(g) \leq t \leq {\Greekmath 010D}(g) + \bar{e}.
\end{align*}
\end{namedassumption}
\end{comment}
\begin{comment}
\begin{namedassumption}{SC-stg}[Synthetic Control in Staggered Design] \label{asm:SC-staggered}
For each eventually treated group $g$, there exists a vector of weight functions $w^g \equiv (w^g_{g'},g' \in \mathcal{D}_{g,\bar{e}})$ whose elements sum to one and satisfy that for almost all $x \in \mathcal{X}$,
\begin{align}
\mathbb{E}[Y_{t}(\infty)|G=g,X=x] = \sum_{g' \in \mathcal{D}_{g,\bar{e}}} w^g_{g'}(x) \mathbb{E}[Y_{t}(\infty)|G=g',X=x], t =1,\ldots,{\Greekmath 010D}(g)+\bar{e}. \label{eqn:w-id-staggered}
\end{align}
Furthermore, ${\Greekmath 010D}(g) \geq |\mathcal{D}_{g,\bar{e}}| \geq 2$, for all $g$, where $| \cdot |$ denotes the cardinality of a set.
\end{namedassumption}
\end{comment}
Similar to \cite{callaway2021difference}, Assumption \ref{asm:PT-staggered}
assumes that the parallel trends assumption holds only starting from the
post-treatment periods. For a given $g$ and $e$, the identification of $
\func{ATT}(g,{\Greekmath 010D} (g)+e)$ follows a logic analogous to the single treated
unit and single treatment period framework studied in Section \ref{sec:setup}
. The adjustments involve shifting the treatment period from $\mathcal{T}$
to ${\Greekmath 010D} (g)+e$ and the pre-treatment periods from $\{1,\ldots ,\mathcal{T}
-1\}$ to $\{1,\ldots ,{\Greekmath 010D} (g)-1\}$. Additionally, the set of control
groups is updated from $\{2,\ldots,N_{G}+1\}$ to $\mathcal{D}_{g,\bar{e}}$.
The condition ${\Greekmath 010D} (g)\geq |\mathcal{D}_{g,\bar{e}}|$ ensures that there are a sufficient number of pre-treatment periods for each group to identify
the weights. This is similar to the setup in \cite{ben2022synthetic} in that
all eventually treated groups remain untreated for some time before
receiving treatment.
The moment function for estimating $\func{ATT}(g,{\Greekmath 010D} (g)+e),e\geq 0,$ is
defined as
\begin{align*}
& {\Greekmath 011E} ^{g,e}(S;m_{\Delta ,g,{\Greekmath 010D} (g)+e},p,w^{g};{\Greekmath 0119} _{g}) \\
=& \frac{1}{{\Greekmath 0119} _{g}}\left( \mathcal{G}_{g}-\sum_{g^{\prime }\in \mathcal{D}
_{g,\bar{e}}}w_{g^{\prime }}^{g}(X)\frac{p_{1}(X)}{p_{g^{\prime }}(X)}
\mathcal{G}_{g^{\prime }}\right) (Y_{{\Greekmath 010D} (g)+e}-Y_{{\Greekmath 010D}
(g)-1}-m_{\Delta ,g,{\Greekmath 010D} (g)+e}(X)),
\end{align*}
where ${\Greekmath 0119} _{g}\equiv \mathbb{P}(G=g)$ and $m_{\Delta ,g,{\Greekmath 010D} (g)+e}\left(
x\right) \equiv \mathbb{E}[Y_{{\Greekmath 010D} (g)+e}-Y_{{\Greekmath 010D} (g)-1}|G\in \mathcal{D}
_{g,\bar{e}},X=x]$. The dependence of $m_{\Delta ,g,{\Greekmath 010D} (g)+e}\left(
x\right) $, $w_{g^{\prime }}^{g}\left( \cdot \right) $ and ${\Greekmath 011E} ^{g,e}$ on $
\bar{e}$ is suppressed.
\begin{corollary}
\label{cor:staggered} Let Assumptions \ref{asm:panel}, \ref{asm:overlap},
and \ref{asm:NA-staggered} hold. \
\begin{enumerate}
\item If Assumption \ref{asm:PT-staggered} holds and $0\leq e\leq \bar{e}
\leq \mathcal{T}-{\Greekmath 010D} (g)$, then
\begin{equation*}
\func{ATT}(g,{\Greekmath 010D} (g)+e)=\mathbb{E}[{\Greekmath 011E} ^{g,e}(S;m_{\Delta ,g,{\Greekmath 010D}
(g)+e},\tilde{p},\tilde{w}^{g};{\Greekmath 0119} _{g})]=\mathbb{E}[{\Greekmath 011E} ^{g,e}(S;\tilde{m}
_{\Delta ,g,{\Greekmath 010D} (g)+e},p,\tilde{w}^{g};{\Greekmath 0119} _{g})],
\end{equation*}
for any weight functions $\Tilde{w}^{g}$ satisfying $\sum_{g^{\prime }\in
\mathcal{D}_{g,\bar{e}}}\tilde{w}_{g^{\prime }}^{g}(x)=1$ for all $x\in
\mathcal{X}$, any non-zero functions $\Tilde{p}$, and any functions $\tilde{m
}_{\Delta ,g,{\Greekmath 010D} (g)+e}$.
\item If Assumption \ref{asm:SC-staggered} holds and $0\leq e\leq \bar{e}
\leq \mathcal{T}-{\Greekmath 010D} (g)$, then
\begin{equation*}
\func{ATT}(g,{\Greekmath 010D} (g)+e)=\mathbb{E}[{\Greekmath 011E} ^{g,e}(S;\tilde{m}_{\Delta
,g,{\Greekmath 010D} (g)+e},p,w^{g};{\Greekmath 0119} _{g})],
\end{equation*}
for any function $\tilde{m}_{\Delta ,g,{\Greekmath 010D} (g)+e}$, where $w^{g}$ is the
vector of weight functions satisfying (\ref{eqn:w-id-staggered}).
\end{enumerate}
\end{corollary}
The semiparametric estimation and bootstrap inference proceed as in Section
\ref{sec:asymptotics} and are omitted here for brevity. The results under
the staggered adoption design can be extended to the repeated
cross-sectional setting in a manner analogous to Section \ref{sec:rc}.
\section{Empirical study}
\label{sec:empirical}
In this section, we apply our method to study the impact of minimum wage on
family income, utilizing the natural experiment of the 2003 minimum wage
increase in Alaska, as examined by \cite{gunsilius2023distributional}. The
effect of minimum wage policies is of significant importance for labor
market dynamics and poverty alleviation strategies, making it a key area of
concern for economists and policymakers. However, the impact is complex and
not clear \textit{a priori}, often involving a trade-off between higher
earnings for certain low-wage workers and potential negative outcomes for
others, such as job losses, reduced work hours, or inflation pressures,
which may diminish the overall gains. Minimum wage is also the focus of \cite
{card1994minimum}, a seminal paper often regarded as one of the most
influential studies in the DiD literature.
\cite{gunsilius2023distributional} used a subset of the data provided by
\cite{dube2019minimum}, which was derived from individual-level data in the
Current Population Survey (CPS). \cite{dube2019minimum} employed a two-way
(state and time) fixed effects regression, along with other specifications
that included additional controls and trends. This approach can be viewed as
a parametric specification of the parallel trends assumption in Assumption
\ref{asm:PT}, though such a parametric form is stronger than necessary for
our purpose. In contrast, \cite{gunsilius2023distributional} proposed the
distributional synthetic control method and used it to construct a synthetic
Alaska by matching state-level quantile curves of family income. Unlike
their approach, our method allows for consistent estimation and valid
inference of the ATT, regardless of whether the parallel trends or synthetic
control assumption holds.\footnote{\cite{dube2019minimum} primarily focused
on the poverty-reducing effects and estimated the unconditional quantile
partial effect of minimum wage increases on family income for families at
lower income quantiles. Similarly, \cite{gunsilius2023distributional}
explored shifts across the entire family income distribution to illustrate
the distributional synthetic control method. To maintain clarity and focus
in our application, we restrict our discussion to the effect on the mean
family income.}
We provide a brief description of the data and variables used in the
analysis. In the CPS, different households are rotated in and out of the
survey at different time intervals. As a result, similar to \cite
{gunsilius2023distributional}, we treat the dataset as repeated
cross-sectional data. The dataset covers a six-year period, from 1998 to
2003, during which the treatment group, Alaska, increased its minimum wage
from \$5.65 to \$7.15 in the final year. During this period, 33 other states
did not change their minimum wage and thus could potentially serve as
control states. Since our method requires the number of control states not
to exceed the length of the time period, we select six control states based
on the analysis in \cite{gunsilius2023distributional}, namely the states
with the most significant weights in their synthetic control analysis:
Virginia, New Hampshire, Maryland, Utah, Michigan, and Ohio.\footnote{
Specifically, in \cite{gunsilius2023distributional}'s study, the four
control states with the largest weights in the method using quantile curves
were Virginia (0.11), New Hampshire (0.11), Maryland (0.09), and Utah
(0.07), where the value in parentheses represents the weight assigned to
each state in constructing the synthetic control. The four control states
with the largest weights in the method relying on the mixture of cumulative
distributions were Michigan (0.12), Ohio (0.10), Maryland (0.10), and
Virginia (0.07). Combining these, we obtain the six control states.}
We define a household as an individual unit $i$ in the study. The outcome
variable $Y$ is the equivalized family income, defined as
multiples of the federal poverty threshold, as in \cite{dube2019minimum}.
The grouping variable $G$ represents the state in which the household
resides. We aim to maintain a set of covariates comparable to those in \cite
{dube2019minimum}, incorporating one continuous variable (age) and two
discrete variables (education level and number of children).\footnote{
Given that the household is the unit of analysis in our study, we exclude
person-level covariates such as gender, race, and ethnicity, which were
included in \cite{dube2019minimum}. Additionally, while \cite
{dube2019minimum} included family size as a covariate, we omit it here due
to its high correlation with the number of children, with a correlation
coefficient of 0.75 in the dataset.} To aggregate individual education
levels at the household level, we count the number of individuals with some
college education and categorize households into three levels: 0, 1, and 2
or more. For the number of children, we similarly consider three categories:
0, 1, and 2 or more. These categorizations yield a relatively balanced
sample size across the nine resulting subgroups, with the number of
observations ranging from 4,000 to 12,000. This exercise demonstrates that
the minimal group size can be sufficiently substantial to implement our
method in empirically relevant datasets.
We estimate conditional expectations using local linear regression. Bandwidths are chosen by undersmoothing data‑driven, mean‑squared‑error (MSE)–optimal bandwidths.\footnote{For the nonparametric outcome regression, we first compute the MSE‑optimal bandwidth $h_{\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{cct}}$ from a local polynomial regression of the outcome on age at the median age, using the \texttt{nprobust} package \citep{calonico2019nprobust}. We then set our bandwidth to $n^{1/5-1/3.5} \times h_{\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{cct}}$. Because the MSE‑optimal bandwidth is of rate $n^{-1/5}$, multiplying by $n^{1/5-1/3.5}$ yields an undersmoothed bandwidth of order $n^{-1/3.5}$, which is $o(n^{-1/4})$, as required by our theory (cf. the undersmoothing adjustment in \cite{shen2016distributional}). We select the propensity‑score bandwidth in the same way. Other reasonable choices of the undersmoothing power close to $1/3.5$ give similar results.} For cross‑fitting, we use $L = 2$ folds.
\begin{comment}
\begin{table}[htbp]
\centering
\begin{tabular}{lccc}
\toprule
& \multicolumn{3}{c}{Number of Children} \\
\cmidrule(lr){2-4}
Education & 0 Children & 1 Child & 2+ Children \\
\midrule
\multirow{2}{*}{$0$} & $-0.636$ & $1.737$ & $0.157$ \\
& $(-1.949, 1.287)$ & $(-0.023, 4.291)$ & $(-1.310, 1.955)$ \\ \hline
\multirow{2}{*}{$1$} & $0.819$ & $1.671$ & $0.263$ \\
& $(-1.035, 2.947)$ & $(-0.474, 4.714)$ & $(-1.338, 2.197)$ \\ \hline
\multirow{2}{*}{$2+$} & $-0.838$ & $-0.451$ & $-1.194$ \\
& $(-5.667, 2.471)$ & $(-4.563, 2.668)$ & $(-3.140, 1.219)$ \\
\bottomrule
\end{tabular}
\caption{Estimated ATT from the 2003 Minimum Wage Increase in Alaska}
\label{tab:empirical_results}
\caption*{\footnotesize The table reports the estimated ATT of equivalized family income for nine subgroups, defined by education level and number of children. Education refers to the number of individuals in the household with some college education. Multiplier bootstrap confidence intervals at the 95\% level, constructed using 500 bootstrap replications, are presented in parentheses below each point estimate.}
\end{table}
\end{comment}
Table \ref{tab:empirical_results} reports the estimated ATTs along with the
95\% confidence intervals obtained via the multiplier bootstrap method. In
line with the findings of \cite{gunsilius2023distributional}, our results
indicate no statistically significant immediate effect on family income following the minimum wage increase in Alaska. Several factors may contribute to this outcome. Firms facing higher labor costs might respond by
reducing working hours or adjusting non-wage benefits to offset the
increased wage expenses. In competitive labor markets, the transmission of
higher minimum wages to overall family income can be limited by substitution
effects, where employers may prefer to replace low-wage workers with more
experienced employees.
Due to the double robustness of our method, our findings provide an
effective shield that protects the null effect reported by
\citet{gunsilius2023distributional} against potential violations of the
underlying identification assumptions. Specifically, our results remain
valid as long as either the parallel trends assumption holds for family
income in the absence of the minimum wage change or the synthetic control
structure accurately captures counterfactual income dynamics across states,
but not necessarily both.
\section{Simulation} \label{sec:simulation}
We conduct simulations based on panel-data DGPs calibrated to the empirical study in Section \ref{sec:empirical}. We denote the observed data from the empirical study by $(\{Y_{i,t}^*\}_{t=1}^{\mathcal{T}}, G_i^*, X_i^*)$, to distinguish them from the variables used in the simulation study. The panel matrix of $Y_{i,t}^*$ is imputed from the repeated cross-sectional CPS data using a low-rank matrix completion method. The covariate $X_i^*$ is normalized to lie between 0 and 1.
We set $\mathcal{T} = N_G = 6$ and $n = 1000,2000,3000$. The marginal distribution of $X$ is taken to be a discrete uniform distribution over an equally spaced grid of 101 points on $[0,1]$,\footnote{This ratio of the number of covariate grid points to the sample size is similar to that in the empirical study.} and the conditional distribution $G|X$ is generated from a multinomial logit model estimated from the empirical data. The treated potential outcome is set to
\begin{align*}
Y_{i,t}(1) = Y_{i,t}(0) + \frac{\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{sd}(\Delta Y_{i,T}(0))}{\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{sd}(\sin(2{\Greekmath 0119} X_i))} \sin(2{\Greekmath 0119} X_i) + 1,
\end{align*}
so that the treatment effect is heterogeneous and its variation is comparable to the untreated outcome trend.\footnote{While the unconditional mean of the sinusoidal term is zero, the true ATT is different from 1 because the conditional distribution of $X_i$ given $G=1$ is not uniform over the grid points on $[0,1]$ (cf. Table \ref{tab:sim}).} Below, we set out three data-generating processes for the untreated potential outcome $Y_{i,t}(0)$: DGP1 satisfies Assumption \ref{asm:SC}, DGP2 satisfies Assumption \ref{asm:PT}, and DGP3 satisfies both.
\paragraph{DGP1} For $g \geq 2$, generate the untreated potential outcome according to
\begin{align*}
Y_{i,t}(0) = {\Greekmath 0115}(G_i, X_i)' F_t + {\Greekmath 010B}_{G_i} + {\Greekmath 010E}_t + {\Greekmath 0122}_{i,t}.
\end{align*}
The variables are generated as follows. Let ${\Greekmath 010B}_g = \frac{1}{\mathcal{T}-1} \sum_{t=1}^{\mathcal{T}-1} \bar{Y}_{g,t}^*$ be the sample average of outcomes for each control group. Let ${\Greekmath 010E}_t = \bar{Y}_{\cdot,t}^* = \frac{1}{n} \sum_{i=1}^{n} Y_{i,t}^*$ be the sample average outcome at each time period. Denote $\tilde{Y}_{i,t}^*$ as the double-demeaned outcome:
\begin{align*}
\tilde{Y}_{i,t}^* = Y_{i,t}^* - \frac{1}{n} \sum_{i=1}^{n} Y_{i,t}^* - \frac{1}{\mathcal{T}-1} \sum_{t=1}^{\mathcal{T}-1} Y_{i,t}^* + \frac{1}{n(\mathcal{T}-1)} \sum_{i=1}^{n} \sum_{t=1}^{\mathcal{T}-1} Y_{i,t}^*.
\end{align*}
We estimate a two-factor model on $\tilde{Y}_{i,t}^*$ in the control groups to obtain the loadings ${\Greekmath 0115}_i^*$ and factors $F_t$.
The loading functions ${\Greekmath 0115}(g,x)$ for the control groups are obtained by regressing ${\Greekmath 0115}_i^*$ on $G_i^*$ , $X_i^*$, their interactions, and squared terms.
We define the synthetic control weights as
\begin{align*}
w_g(x) = 0.2 + 0.8 \cdot \frac{\exp\left[(g - N_G - 1) x\right]}{\sum_{g'=2}^{N_G+1} \exp\left[(g' - N_G - 1) x\right]}, \quad g = 2, \dots, N_G+1.
\end{align*}
This guarantees that each weight is at least 0.2 and varies sufficiently with $x$. The loading function for the treatment group is then constructed as the weighted average:
\begin{align*}
{\Greekmath 0115}(1, x) = \sum_{g=2}^{N_G+1} w_g(x) \cdot {\Greekmath 0115}(g, x).
\end{align*}
Similarly, the group fixed effect for the treatment group is constructed as:
\begin{align*}
{\Greekmath 010B}_1 = \sum_{g=2}^{N_G+1} w_g(x) \cdot {\Greekmath 010B}_g.
\end{align*}
Finally, the error terms ${\Greekmath 0122}_{i,t}$ are serially independent and, for each period $t$, are iid draws from a mean‑zero normal distribution whose variance is estimated from the residuals of the factor model for $\tilde{Y}_{i,t}^{*}$.
\paragraph{DGP2}
We generate the untreated potential outcome as
\begin{align*}
Y_{i,t}(0)={\Greekmath 0116}_t(G_i,X_i)+{\Greekmath 010B}_{G_i}+{\Greekmath 010E}_t+{\Greekmath 0122}_{i,t}.
\end{align*}
Let ${\Greekmath 010B}_g=\frac{1}{\mathcal{T}-1}\sum_{t=1}^{\mathcal{T}-1}\bar{Y}_{g,t}^*$ denote the pre-treatment average outcome for group $g$, and generate ${\Greekmath 010E}_t$ in the same way as in DGP1. In the pre-treatment periods $1\le t\le \mathcal{T}-1$, set ${\Greekmath 0116}_t(g,x)=0$ for all control groups $g\ge2$; for the treatment group, estimate ${\Greekmath 0116}_t(1,x)$ by a quadratic regression of $\tilde{Y}_{i,t}^*$ on $X_i^*$ using the treatment-group data at period $t$. In the treatment period, set ${\Greekmath 0116}_{\mathcal{T}}(g,x)={\Greekmath 0116}_{\mathcal{T}-1}(g,x)+h(x)$ for all groups, where $h(x)$ is estimated by a quadratic regression of $\tilde{Y}_{i,\mathcal{T}}^*-\tilde{Y}_{i,\mathcal{T}-1}^*$ on $X_i^*$ using the control-group data. The error terms ${\Greekmath 0122}_{i,t}$ are generated in the same way as in DGP1. In this DGP, Assumption \ref{asm:PT} holds, whereas Assumption \ref{asm:SC} does not, because the weighted average of the zero values ${\Greekmath 0116}_t(g,x)$ for the control groups will always be zero and cannot replicate ${\Greekmath 0116}_t(1,x)$.
\paragraph{DGP3} All variables are generated exactly as in DGP1, except that the loading function ${\Greekmath 0115}$ is now estimated by regressing ${\Greekmath 0115}_i^{*}$ solely on $X_i^{*}$ and its square. Consequently, the loading function no longer depends on group assignment, and Assumption \ref{asm:PT} is satisfied in addition to Assumption \ref{asm:SC}.
The simulation results are reported in Table \ref{tab:sim}. Across all three DGPs, our procedure performs as predicted by theory: the bias is negligible relative to the standard deviation, the standard deviation decreases at the expected $1/\sqrt{n}$ rate, and coverage aligns with the nominal level. The resulting confidence intervals are of reasonable length, with power improving as the sample size increases. We select the bandwidth as $n^{-1/3.5}$, and alternative undersmoothing bandwidths yield similarly robust performance.
\begin{comment}
\begin{table}[htbp]
\centering
\begin{tabular}{lccrcccc}
\toprule
& True ATT & $n$ & Bias & SD & Coverage & CI Length \\
\midrule
\multirow{3}{*}{DGP 1} & \multirow{3}{*}{1.083} & 1000 & -0.002 & 0.085 & 0.952 & 0.469 \\
& & 2000 & 0.004 & 0.058 & 0.936 & 0.270 \\
& & 3000 & 0.007 & 0.046 & 0.920 & 0.210 \\ \hline
\multirow{3}{*}{DGP 2} & \multirow{3}{*}{1.477} & 1000 & -0.025 & 0.418 & 0.959 & 2.270 \\
& & 2000 & -0.002 & 0.287 & 0.927 & 1.310 \\
& & 3000 & 0.011 & 0.220 & 0.945 & 1.016 \\ \hline
\multirow{3}{*}{DGP 3} & \multirow{3}{*}{1.080} & 1000 & -0.007 & 0.082 & 0.958 & 0.459 \\
& & 2000 & -0.000 & 0.058 & 0.918 & 0.265 \\
& & 3000 & 0.002 & 0.044 & 0.930 & 0.206 \\
\bottomrule
\end{tabular}
\caption{Simulation results across DGPs and sample sizes}
\label{tab:sim}
\caption*{\footnotesize The table summarizes the simulation results for the three DGPs. The second column presents the true ATT based on 10000 replications. The fourth and fifth columns show the bias and standard deviation of the point estimator. The last two columns report the coverage and the median length of the bootstrap confidence interval at the 95\% nominal level, computed with 500 bootstrap replications. The reported results are based on 1000 simulation replications.}
\end{table}
\end{comment}
\section{Conclusion}
\label{sec:conclusion}
This paper introduces a doubly robust methodology that unifies DiD and
synthetic control approaches, enabling identification, consistent
semiparametric estimation, and valid bootstrap inference of the ATT under
either parallel trends or synthetic control assumptions. The method is
applicable in a wide range of empirical settings, including panel data,
repeated cross-sectional data, and staggered treatment designs. The
framework is suitable for various micro-level datasets, such as
administrative records, survey data, or digital trace data, where meaningful
heterogeneity across groups can be exploited. We recommend that empirical
researchers adopt this approach to mitigate biases stemming from strong
identifying assumptions and to ensure more robust causal inference.
\bibliographystyle{chicago}
\bibliography{DiD_SC}