Difference-in-Differences Meets Synthetic Control: Doubly Robust Identification and Estimation

Difference-in-Differences Meets Synthetic Control: Doubly Robust Identification and Estimation

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\ 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 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\ 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 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 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 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 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) introduces the panel model setup, along with the identification assumptions and the doubly robust identification result. Section (ref) establishes the semiparametric estimation theory and the multiplier bootstrap theory separately under the parallel trends and synthetic control assumptions. Sections (ref) and (ref) extend the analysis to repeated cross-sectional data and staggered treatment designs, respectively. Sections (ref) and (ref) present the empirical and simulation studies. Section (ref) 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 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 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, robbins2017framework examined the impact of region-level crime interventions using individual-level data, whereas chen2020distributional introduced a distributional synthetic control method based on a quantile factor model. More recently, nonparametric approaches, such as those proposed by gunsilius2023distributional and chen2023group, have been developed to study the synthetic matching of the entire outcome distribution by drawing on the changes-in-changes method from athey2006identification. While gunsilius2023distributional 's method applies to short-panel settings, 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 athey2021matrix, the synthetic DiD method arkhangelsky2021synthetic, and the augmented SC method 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).

Setup and identification

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).} 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).

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 $Y_{t}\equiv \mathcal{G}_{1}Y_{t}(1)+(1-\mathcal{G}_{1})Y_{t}(0)$.

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

The target parameter is

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 $\mathcal{X}$.

Under Assumption (ref), 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 $\{i:G_{i}=g\}=\{i:\mathcal{G}_{gi}=1\}$. \

Identification assumptions

We introduce the following three identification assumptions, with the understanding that they need not all be imposed at once.

Assumption (ref) 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 abadie2005semiparametric.

Assumption (ref) 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) with a factor model structure, as in the conventional synthetic control literature:

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) holds if there exist weighting functions $w_{g}\left( \cdot \right) $ such that

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) takes a different approach from the distributional synthetic control framework in 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 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 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 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) and (ref) reflect two different approaches to identification. Assumption (ref) relies on a strong cross-sectional relationship among groups, though this relationship needs only to hold over a single period. In contrast, Assumption (ref) allows for a weaker cross-sectional relationship but requires it to persist over a longer duration. Importantly, neither Assumption (ref) implies Assumption (ref), 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.

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). Our identification strategy is to construct a moment function whose mean is the target causal parameter ${\Greekmath 0112} $. We propose the following moment function:

which can be equivalently represented as

Before showing that ${\Greekmath 011E} \left( \cdot \right) $ is unbiased for ${\Greekmath 0112}$ under either Assumption (ref) or Assumption (ref), 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)) 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

Here, ((ref)) 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

which is the expected change in the outcome of interest from time $\mathcal{T }-1$ to $\mathcal{T}$ for the control groups. Then

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

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.

Theorem (ref) 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 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:

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} $.

Comparison with existing methods in the case without covariates

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

The simple average estimator based on the above ${\Greekmath 011E} $ is then given by

where $\bar{Y}_{g,t}$ is the average of the outcomes for individuals in group $g$ at time period $t.$

Building on shen2023same's formulation (specifically, their equation (6)) of the synthetic DiD estimator studied by 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

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 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)) becomes our estimator in ((ref)). Theorem 1 of 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.

Estimation and inference

The identification result in Theorem (ref) establishes that estimators based on ${\Greekmath 011E}$ are consistent for ${\Greekmath 0112}$ as long as either Assumption (ref) or Assumption (ref) 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.

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

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), when only Assumption (ref) holds, the synthetic weight $w$ may not be well-defined, meaning that there may not exist any $w$ such that ((ref)) 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)).

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.

The asymptotic variance $V_{\mathrm{PT}}$ in ((ref)) 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 sant2020doubly. More specifically, in the $2\times 2$ design with a single control group $g$, the efficient influence function for the ATT is

Equation ((ref)) 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 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 sant2020doubly becomes

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)) 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) aligns with the framework of double/debiased machine learning (DML) as discussed in chernozhukov2018DML. Under Assumption (ref), 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)(i), which shows that the nuisance weights play a secondary role under Assumption (ref).

\paragraph{Asymptotic theory under synthetic control}

Deriving the asymptotic distribution of $\hat{{\Greekmath 0112}}$ under Assumption (ref) 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

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)(ii).}

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), the weights can be identified via the equation:

After rearranging terms, we obtain that $M\bm{w}_{0}=m_{1}$, where

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

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:

Since ${\Greekmath 011E} $ is not Neyman orthogonal under Assumption (ref), the required rate of convergence for each nuisance function estimator, specified in the first condition of Theorem (ref), has been strengthened to $o_{p}\left( n^{-1/4}\right) $ compared to Theorem (ref). This rate is typical for first-step nonparametric estimators newey1994asymptotic,chen2003estimation. The second condition of Theorem (ref) 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, kong2010uniform.

The asymptotic properties of our estimator fall within the broader framework of semiparametric two-step estimation theory chen2003estimation. Our main contribution in Theorem (ref) is to derive the adjustment to the asymptotic variance to account for the nuisance estimators, which corresponds to Condition (2.6) in 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) 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 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).

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)) to ((ref)), 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)), 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

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.

This multiplier bootstrap approach differs from the nonparametric bootstrap, where the bootstrap weights $(W_{1},\ldots,W_{n})$ follow a multinomial distribution. Assumption (ref) 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 $(\mathcal{W}_{n},\mathcal{S}_{n})$.

In Theorem (ref), 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). 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 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), 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), 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

Finally, we emphasize that Theorems (ref), (ref), and (ref) 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)) and ((ref)), 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.

Repeated cross-sectional data

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 meyer1995workers and car accidents cohen2003effects, longitudinal analyses of fixed-age cohorts over extended periods corak2001death, and nationally representative surveys, such as the Current Population Survey acemoglu2001consequences and the Canadian General Social Surveys 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

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.

Under the repeated cross-sectional sample scheme in Assumption (ref), $T_{i}$ 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) aligns with the time invariance conditions commonly found in the DiD literature, such as Assumption 3.3 in abadie2005semiparametric. Under this assumption, the joint distribution of $(Y,G,X,T)$ can be expressed through the following mixture representation:

Similar to the repeated cross-sectional settings discussed in sant2020doubly and callaway2021difference, Assumption (ref) excludes compositional changes, that is, changes in the distribution of $G$ and $X$ over time. Recent work by 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),

where $m_{g,t}\left( x\right) =\mathbb{E}[Y|G=g,X=x,\mathfrak{T}_{t}=1].$ Similarly, under Assumption (ref),

where $m_{G\neq 1,t}\left( x\right) =\mathbb{E}[Y|G\neq 1,X=x,\mathfrak{T} _{t}=1].$ Define

We will still maintain Assumption (ref) or (ref). To facilitate interpretation, we rewrite them in an equivalent form. For Assumption (ref), we express it as:

or equivalently

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), we express it as

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:

where the factor $1/{\Greekmath 0115} _{t}$ on both sides cancels out. Similar to ((ref)), 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

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

For estimation, we consider the same cross-fitting method as in the panel setting. The estimator for ${\Greekmath 0112} $ is

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}$.

Similar to the panel setting, we construct the bootstrap estimator as

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.

Staggered treatment adoption

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.

The causal effects of interest are the group-level ATTs in post-treatment periods:

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:

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 ${\Greekmath 010D} (g)+e$:

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 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 $e\in \{0,1,\ldots ,\bar{e}\}$.

Similar to callaway2021difference, Assumption (ref) 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) . 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 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

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.

The semiparametric estimation and bootstrap inference proceed as in Section (ref) 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).

Empirical study

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 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 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 card1994minimum, a seminal paper often regarded as one of the most influential studies in the DiD literature.

gunsilius2023distributional used a subset of the data provided by dube2019minimum, which was derived from individual-level data in the Current Population Survey (CPS). 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), though such a parametric form is stronger than necessary for our purpose. In contrast, 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{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, 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 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 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 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 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 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 dube2019minimum. Additionally, while 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 nprobust package 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 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.

Table (ref) reports the estimated ATTs along with the 95% confidence intervals obtained via the multiplier bootstrap method. In line with the findings of 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 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.

Simulation

We conduct simulations based on panel-data DGPs calibrated to the empirical study in Section (ref). 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

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)).} Below, we set out three data-generating processes for the untreated potential outcome $Y_{i,t}(0)$: DGP1 satisfies Assumption (ref), DGP2 satisfies Assumption (ref), and DGP3 satisfies both.

\paragraph{DGP1} For $g \geq 2$, generate the untreated potential outcome according to

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:

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

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:

Similarly, the group fixed effect for the treatment group is constructed as:

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

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) holds, whereas Assumption (ref) 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) is satisfied in addition to Assumption (ref).

The simulation results are reported in Table (ref). 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.

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.