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Graph-Regularized Synthetic Control for Stable Counterfactual Estimation under Donor Redundancy
\maketitle
\begin{abstract}
Synthetic control methods estimate counterfactual outcomes by weighting untreated donor units to match a treated unit before treatment. In recent applications, however, the donor pool can be large, highly correlated, and short in pre-treatment history, yielding many near-equivalent solutions with unstable and difficult-to-interpret donor weights. We propose Graph-Regularized Synthetic Control (Graph-SC), a convex estimator that augments the classical pre-treatment matching objective with a graph Laplacian penalty. The graph encodes similarity among donor trajectories, encouraging coherent weights over structurally related donors while preserving the transparency of synthetic controls. We provide theoretical results characterizing the bias-variance tradeoff induced by graph regularization and establish efficient convergence guarantees with an accelerated proximal gradient method. Across a controlled simulation under donor redundancy, the California Proposition 99 benchmark, and observational counterfactual analyses on the Waymo Open Motion Dataset, Graph-SC improves weight stability, reduces sensitivity to redundant donor pools, and maintains competitive pre- and post-treatment accuracy relative to classical and regularized synthetic control baselines. Our results suggest that graph regularization is most valuable not as a universal accuracy booster, but as a principled mechanism for selecting stable and interpretable counterfactuals from near-equivalent synthetic controls with comparable pre-treatment fit.
\end{abstract}
\section{Introduction}
Synthetic control (SC) is arguably the most important
Innovation in the policy evaluation literature \cite{athey2017state}. It estimates the counterfactual trajectory of a treated unit (e.g., a region or individual exposed to an intervention) by forming a weighted combination of untreated donors (e.g., comparable regions or individuals not exposed to the intervention) whose pre-intervention behavior matches the treated unit \cite{abadie2003economic,abadie2010synthetic}. Its appeal lies in its interpretability, since the counterfactual is constructed as an explicit mixture of real units, and in its simplicity, as treatment effects are computed directly from post-intervention discrepancies.
A fundamental challenge arises in modern, data-rich applications with large and highly correlated donor pools. When many donors exhibit similar pre-intervention behavior, the weight learning problem admits multiple near equivalent solutions that achieve indistinguishable pre-fit. As a result, the estimation becomes ill-conditioned, with learned weights that are overly sensitive to small perturbations such as noise, resampling, window selection, or feature scaling. This sensitivity propagates to the counterfactual, yielding high variance estimates and unstable treatment effects. The central question motivating this work is whether counterfactual construction can be stabilized without sacrificing the transparency of donor weight mixtures.
We propose \textbf{Graph Synthetic Control (Graph-SC)}, a convex estimator that incorporates pre-intervention donor structure into synthetic control through a learned similarity graph. The donor graph is constructed from pre-intervention features, and the weight vector is regularized via a Laplacian quadratic form. Intuitively, donors who behave similarly before treatment should receive similar weights. Formally, the graph regularizer couples weights along graph edges, selecting stable solutions among many near equivalent fits while preserving interpretability. Graph-SC further combines this structural prior with sparsity, enforced via an $\ell_1$ penalty, yielding stable donor mixtures and a well-conditioned optimization problem. Our Contribution can be summarized as follows.
\begin{itemize}
\item \textbf{Estimator.} We introduce Graph-SC, a graph-regularized synthetic control method that combines pre-intervention matching, $\ell_1$ sparsity, and Laplacian smoothing on a donor similarity graph learned from pre-intervention data.
\item \textbf{Theory.}
We characterize how graph regularization affects stability through a bias-variance tradeoff. We derive finite sample upper bounds on estimation error that quantify variance reduction from graph smoothing, and a complementary lower bound showing that when the learned graph is misspecified, the induced bias is unavoidable. These results delineate when Graph-SC outperforms classical synthetic control.
\item \textbf{Computation.} We show Graph-SC is efficiently solvable by (accelerated) proximal gradient methods \citep{beck2009fast} and establish convergence rates specialized to the Graph-SC composite objective.
\end{itemize}
\section{Related Work and Setting}
\subsection{Related Work}
\label{subsec:related_work}
\paragraph{Synthetic control and panel counterfactual.}
Classical SC was introduced for comparative case studies in economics and policy evaluation \cite{abadie2003economic,abadie2010synthetic}. Subsequent work connected SC to broader panel estimators and improved robustness via reweighting, including augmented SC \cite{ben2021augmented}, synthetic difference-in-differences \cite{arkhangelsky2021synthetic}, interactive fixed-effect imputation \cite{xu2017generalized}, and low-rank/matrix-denoising approaches for noisy panels \cite{amjad2018robust}. These methods primarily regularize the structure of the outcome panel, whereas our approach is complementary, regularizing the donor weights directly. By learning a similarity graph from pre-intervention data and applying graph-based smoothing to the weights, we address instability arising from large, highly correlated donor pools.
\paragraph{Graph Regularization}
Regularized variants of SC modify the original constraints or objective to improve performance with many donors, including elastic-net viewpoints and penalized SCM \cite{abadie2021penalized}. Graph-structured regularization, including Laplacian-based smoothing and network-guided penalties, is well studied in regression and clustering \citep{li2008network, hallac2015network}. Graph-SC adapts these ideas to counterfactual construction by encoding a pre-intervention similarity prior over donors through a graph. This induces structured, group-aware donor selection, improves stability, and preserves the counterfactual as an interpretable mixture of real donors.
\subsection{Problem setup}
\label{subsec:setup}
We consider a standard comparative case study with one treated unit (index $1$) and a donor pool of $J$ untreated units (indices $2,\dots,J{+}1$). Time is indexed by $t\in\{1,\dots,T\}$, with an intervention at $T_0$ (pre-intervention $t\le T_0$, post-intervention $t>T_0$). Let $y_{it}(0)$ and $y_{it}(1)$ denote the potential outcomes without and with treatment.
\begin{definition}
\label{def:sc}
Classical synthetic control (SC) \cite{abadie2003economic} estimates donor weights as a convex combination that matches pre-intervention features:
\begin{equation}
\label{eq:sc}
\widehat w_{\mathrm{SC}}
\in
\arg\min_{w\in\mathbb{R}^J}
\frac12\|x_1-X_0 w\|_2^2
\quad
\text{s.t.}\quad
w\ge 0,\;\mathbf{1}^\top w=1.
\end{equation}
The counterfactual is $\widehat y_{1t}(0)=Y_{0t}^\top \widehat w_{\mathrm{SC}}$ for $t>T_0$.
\end{definition}
Classical SC is appealing because it is simple and transparent. However, when $J$ is large and donors are correlated, there may be many distinct weight vectors that achieve essentially the same pre-intervention fit, which makes the counterfactual sensitive to noise and modeling choices.
\begin{proposition}
\label{prop:nonunique}
Suppose there exists a nonzero vector $v\in\mathbb{R}^J$ such that $X_0 v=0$ and $\mathbf{1}^\top v=0$. If $\widehat w$ is feasible for \eqref{eq:sc}, then for all sufficiently small $t\in\mathbb{R}$ such that $\widehat w + t v\ge 0$, the perturbed weight vector $\widehat w + t v$ is also feasible and attains the same objective value:
\[
\|x_1-X_0(\widehat w+t v)\|_2^2 = \|x_1-X_0\widehat w\|_2^2.
\]
Hence, the SC solution is not unique. The proof is provided in subsection ~\ref{proof:nonunique}.
\end{proposition}
\begin{remark}
\label{rem:illposed}
In practice, donors are often nearly collinear, so making $X_0^\top X_0$ ill-conditioned. This creates nearly flat directions in the SC objective: small pre-intervention noise can induce large changes in $\widehat w_{\mathrm{SC}}$, inflating the variance of post-intervention effects. The rest of our method introduces a structured inductive bias to select among many near-equivalent pre-fit solutions.
\end{remark}
\section{Graph-Regularized Synthetic Control}
\subsection{Learning donor similarity graphs}
\label{:graph_learning}
\begin{definition}
\label{def:graph}
Let the donor pool be indexed by $u\in\{1,\dots,J\}$, corresponding to original units $j=u+1$ with pre-intervention feature vectors $x_{u+1}\in\mathbb{R}^k$. Fix a symmetric similarity function $s:\mathbb{R}^k\times\mathbb{R}^k\to\mathbb{R}$ computed using pre-intervention data only, and a neighborhood size $k_{\mathrm{nn}}\in\mathbb{N}$. Define the similarity matrix $S\in\mathbb{R}^{J\times J}$ by
\[
S_{uv}=
\begin{cases}
s(x_{u+1},x_{v+1}), & u\neq v,\\
-\infty, & u=v.
\end{cases}
\]
For each $u$, let $\mathcal{N}(u):=\mathrm{TopKIndices}(S_{u,:},k_{\mathrm{nn}})$ be the indices of the $k_{\mathrm{nn}}$ largest entries of row $u$. Define a directed $k$NN adjacency $A^{\mathrm{dir}}\in\{0,1\}^{J\times J}$ by $A^{\mathrm{dir}}_{uv}=\mathbf{1}\{v\in\mathcal{N}(u)\}$, then symmetrize to obtain an undirected unweighted adjacency
\[
A \;:=\; \mathbf{1}\{A^{\mathrm{dir}}+(A^{\mathrm{dir}})^\top>0\},
\qquad A_{uu}=0.
\]
Let $d_u=\sum_{v=1}^J A_{uv}$ and $D=\mathrm{diag}(d_1,\dots,d_J)$. The normalized Laplacian is
\[
L \;:=\; I - D^{-1/2} A D^{-1/2},
\]
with the convention $(D^{-1/2})_{uu}=0$ if $d_u=0$.
\end{definition}
\paragraph{Graph Implementation.} \label{rem:graph_complexity}Algorithm~\ref{alg:graph_learning} summarizes the implementation. It first computes pairwise similarities between donors, then forms a $k_{\mathrm{nn}}$-nearest-neighbor graph, and finally symmetrizes it to ensure the downstream regularizer is a positive semidefinite quadratic form. Graph construction is $O(J^2k)$ naively and can be made near-linear in $J$ with details in appendix~\ref{proof:graph_complexity}.
\begin{algorithm}[h]
\caption{Donor Similarity Graph Learning}
\label{alg:graph_learning}
\begin{algorithmic}[1]
\Require Donor pre-intervention features $\{x_j\}_{j=2}^{J+1}$, similarity function $s(\cdot,\cdot)$, neighborhood size $k_{\mathrm{nn}}$
\Ensure Normalized graph Laplacian $L$
\State Initialize $A^{\mathrm{dir}}\gets 0\in\mathbb{R}^{J\times J}$
\State Compute pairwise similarities $S_{uv}\gets s(x_{u+1},x_{v+1})$ for all $u\neq v$, and set $S_{uu}\gets -\infty$
\For{$u=1,\ldots,J$}
\State $\mathcal{N}(u)\gets \mathrm{TopKIndices}(S_{u,:}, k_{\mathrm{nn}})$
\For{each $v\in\mathcal{N}(u)$}
\State $A^{\mathrm{dir}}_{uv}\gets 1$
\EndFor
\EndFor
\State $A \gets \mathbf{1}\{A^{\mathrm{dir}}+(A^{\mathrm{dir}})^\top>0\}$
\State $D \gets \mathrm{diag}(d_1,\ldots,d_J)$, where $d_u=\sum_{v=1}^J A_{uv}$
\State $L \gets I - D^{-1/2} A D^{-1/2}$
\end{algorithmic}
\end{algorithm}
\begin{lemma}[Edge-wise interpretation of Laplacian smoothness]
\label{lem:laplacian_identity}
For the normalized Laplacian $L$ in Definition~\ref{def:graph},
\[
\begin{aligned}
w^\top L w
&= \frac{1}{2}\sum_{u=1}^{J}\sum_{v=1}^{J} A_{uv}
\left(
\frac{w_u}{\sqrt{d_u}}-\frac{w_v}{\sqrt{d_v}}
\right)^2 \\
&= \sum_{(u,v)\in E}
\left(
\frac{w_u}{\sqrt{d_u}}-\frac{w_v}{\sqrt{d_v}}
\right)^2 ,
\end{aligned}
\]
where \(E=\{(u,v):u<v,\ A_{uv}=1\}\).
The proof is provided in appendix~\ref{proof:laplacian_identity}
\end{lemma}
\begin{remark}
\label{rem:graph_prior}
Lemma~\ref{lem:laplacian_identity} shows that $w^\top L w$ penalizes edge-wise difference between weights of similar donors. Therefore, it selects among many pre-fit solutions by favoring donor subnetworks, meaning that if donors are connected in the similarity graph, large weight differences are discouraged.
\end{remark}
\subsection{Graph-Regularized Synthetic Control Estimator}
We incorporate donor structure into synthetic control by introducing a graph-based regularizer that acts as a soft prior: donors that exhibit similar pre-intervention behavior should receive similar weights. This yields a convex estimator that preserves the transparency of weight-based counterfactual construction, while avoiding brittle hard constraints such as exact sum-to-one.
This section addresses the following research question: With large, highly correlated donor pools, can a similarity-induced graph prior reduce the variance of the estimated counterfactual and stabilize treatment effect estimates? Moreover, when does the graph term reduce variance and improve stability, and what bias can it introduce?
\begin{definition}
\label{def:gsc}
Given $x_1$, $X_0$, and Laplacian $L$, define Graph Synthetic Control (Graph-SC) and estimate weights by
\begin{equation}
\label{eq:gsc}
\begin{aligned}
\widehat w
\in \arg\min_{w\in\mathbb{R}^J}
\;&
\frac12\|x_1-X_0 w\|_2^2
+ \lambda_1\|w\|_1 \\
&\quad
+ \frac{\lambda_g}{2} w^\top L w
+ \frac{\lambda_{sum}}{2}(\mathbf{1}^\top w-1)^2 .
\end{aligned}
\end{equation}
with tuning parameters $\lambda_1,\lambda_{g},\lambda_{sum}\ge 0$.
\end{definition}
\begin{assumption}[Similarity graph]
\label{as:graph_connected}
The donor similarity graph is undirected and connected, and $L$ is its Laplacian.
\end{assumption}
\begin{assumption}[Pre-intervention noise model]
\label{as:pre_noise}
There exists a target weight vector $w^\star\in\mathbb{R}^J$ with $w^\star\ge 0$
such that
\[
x_1 = X_0 w^\star + \varepsilon,\qquad \mathbb{E}[\varepsilon]=0,\qquad
\mathbb{E}[\varepsilon\varepsilon^\top]=\sigma_x^2 I_K,
\]
and $\varepsilon$ is sub-Gaussian.
\end{assumption}
\begin{definition}
\label{def:mu}
Define the quadratic regularizer Hessian
\[
Q \;\triangleq\; \lambda_g L + \lambda_{sum} \mathbf{1}\mathbf{1}^\top.
\]
Let $\lambda_2(L)$ denote the second-smallest eigenvalue of $L$.
Under Assumption~\ref{as:graph_connected}, $L$ has eigenvalues
$0=\lambda_1(L)<\lambda_2(L)\le \cdots$.
Define the \emph{effective curvature}
\begin{equation}
\label{eq:mu_def}
\mu \;\triangleq\; \lambda_{\min}(Q)
\;=\; \min\{\lambda_g\lambda_2(L),\;\lambda_{sum} J\}.
\end{equation}
\end{definition}
\paragraph{Bias--variance trade-off.} Variance reduction comes from graph regularization, and bias arises from shrinkage toward graph-smooth weights. We formalize this with an ``optimal'' weight vector $w^\star$ as the target that matches the noiseless pre-intervention features and a population minimizer $\bar w$ defined by replacing $x_1$ with its conditional mean $X_0 w^\star$.
\begin{definition}
\label{def:population_minimizer}
\begin{equation}
\label{eq:population_minimizer}
\begin{aligned}
\bar w
\in \arg\min
\;&
\frac12\|X_0 w^\star - X_0 w\|_2^2
+ \lambda_1 \|w\|_1 \\
&\quad
+ \frac{\lambda_g}{2} w^\top L w
+ \frac{\lambda_{sum}}{2}(\mathbf{1}^\top w - 1)^2 .
\end{aligned}
\end{equation}
The regularization bias is $\bar w - w^\star$.
\end{definition}
\begin{theorem}[Bias and variance bounds]
\label{thm:bias_variance_tradeoff}
Under Assumptions~\ref{as:graph_connected}--\ref{as:pre_noise}, let $\mu$ be as in
Definition~\ref{def:mu}. Then the estimation error admits the decomposition.
\[
\widehat w - w^\star = (\widehat w - \bar w) + (\bar w - w^\star),
\]
with the following bounds.
\smallskip
\noindent\textbf{Variance term upper bound:}
\begin{equation}
\label{eq:estimation_term_bound}
\mathbb{E}\big[\|\widehat w - \bar w\|_2^2\big]
\;\le\;
\frac{\sigma_x^2}{\mu^2}\,\mathrm{tr}(X_0^\top X_0).
\end{equation}
\smallskip
\noindent\textbf{Bias term upper bound:}
There exists a subgradient $\xi\in\partial \| \bar w\|_1$ with $\|\xi\|_\infty\le 1$
such that
\begin{equation}
\label{eq:bias_upper}
\begin{aligned}
\|\bar w - w^\star\|_2
\;\le\;
\frac{1}{\mu}\Big(
&\lambda_1\|\xi\|_2 + \lambda_g\|L w^\star\|_2 + \\&\lambda_{sum}|\mathbf{1}^\top w^\star-1|\sqrt{J}
\Big).
\end{aligned}
\end{equation}
In particular, if $\mathbf{1}^\top w^\star=1$ then the last term vanishes and the bias is controlled by the graph-misspecification measure $\|L w^\star\|_2$.
\smallskip
\noindent\textbf{Bias term lower bound under graph misspecification.}
In the graph-only case ($\lambda_1=0$) with $\mathbf{1}^\top w^\star=1$,
\begin{equation}
\label{eq:bias_lower}
\|\bar w - w^\star\|_2
\;\ge\;
\frac{\lambda_g}{\|X_0^\top X_0 + Q\|_{\mathrm{op}}}\,
\|P_{\mathbf{1}^\perp}(L w^\star)\|_2,
\end{equation}
where $P_{\mathbf{1}^\perp}$ projects onto the subspace orthogonal to $\mathbf{1}$.
The proof is provided in appendix~\ref{proof:bias_variance_tradeoff}
\end{theorem}
\begin{remark}
Graph regularization reduces variance at rate $\mu^{-2}$ via \eqref{eq:estimation_term_bound} but introduces bias proportional to $\lambda_2\|Lw^\star\|_2/\mu$ via \eqref{eq:bias_upper}. Therefore, Graph-SC is most beneficial when $w^\star$ is smooth (i.e., $\|Lw^\star\|_2$ is small) and the donor design is ill-conditioned (i.e., large $\mathrm{tr}(X_0^\top X_0)$), which is the case where classical SC is unstable. Sparse-only ($\lambda_g=0$) can still be unstable when donors are highly correlated (multiple sparse mixtures fit similarly), while graph-only ($\lambda_1=0$) can spread weights over a neighborhood and introduce bias if the similarity graph is misspecified. Therefore, the combined objective balances these effects: $\lambda_1$ controls sparsity and $\lambda_g$ stabilizes and groups along the graph.
\end{remark}
\subsection{Fast optimization for Graph-SC via proximal gradient}
Graph-SC is a convex composite problem of the form $\min_{w\in\mathbb{R}^J}\; F(w)\triangleq f(w)+g(w)$, where $f$ is smooth with Lipschitz gradient and $g$ is proximable.
This places Graph-SC directly in the standard setting of proximal gradient methods (FISTA) \cite{beck2009fast}, and can achieve a convergence rate of \(\mathcal{O}(1/k^2)\). The corresponding setup and proof can be found in appendix ~\ref{proof:conv_gsc_compact}.
\section{Experiments}\label{sec:experiments}
We evaluate Graph-Regularized Synthetic Control (Graph-SC) across three settings: a controlled semi-synthetic benchmark with known treatment effects, the California Proposition 99 policy benchmark, and trajectory-level observational counterfactual analysis using the Waymo Open Motion Dataset (WOMD). The experiments are designed around a central question: when many donor combinations produce similar pre-treatment fit, can graph regularization select a more stable and interpretable synthetic control? We therefore report not only conventional pre-treatment fit and post-treatment discrepancy, but also donor-weight concentration, graph coherence, placebo behavior, and sensitivity to donor-pool perturbations.
\subsection{Semi-Synthetic Study: Donor Redundancy and Bias--Variance Trade-off}\label{subsec:simulation}
We first study a setting where the true counterfactual and treatment effect are known. The purpose is not to mimic a particular empirical dataset, but to isolate the estimator behavior under donor redundancy. We generate donor trajectories from latent smooth clusters so that many donors are individually correlated with the treated unit, and many donor combinations achieve similar pre-treatment fit. This setting creates a synthetic-control Rashomon set: multiple weight vectors have nearly identical pre-treatment loss but can imply different post-treatment counterfactuals.
\noindent\textbf{Data Generation.}
We simulate $N=200$ donor units and one treated unit over $T=70$ periods, with $T_0=50$ pre-treatment and $T_1=20$ post-treatment periods. Donors are assigned to $K=10$ latent clusters. Each cluster trajectory is generated from a low-frequency Fourier basis plus a linear trend, and unit-level trajectories are obtained by adding Gaussian noise. The treated unit is generated as a convex combination of donors from one latent cluster and receives a constant additive treatment effect $\tau=2.0$ after $T_0$.
\noindent\textbf{Estimators.} We compare standard SC, graph-only regularization, sparse-only regularization, and the full Graph-SC objective. The graph is constructed from donor pre-treatment trajectories using $k$-nearest neighbors unless otherwise specified. In this controlled setting, we additionally evaluate recovery of the true donor cluster and stability under donor resampling.
\noindent\textbf{Results.}
Figure~\ref{fig:simulation_results} shows the performance metrics across 200 repetitions, revealing distinct failure modes in the baselines that Graph-SC effectively overcomes. The Graph Regularization Only estimator (Orange) exhibits extreme instability with a variance of approximately 20.24, frequently overestimating the effect magnitude, which confirms that graph smoothness alone is insufficient for conditioning the problem without sparsity. Conversely, the Sparse Regularization Only estimator (Green) successfully reduces variance to approximately 0.44 but suffers from severe shrinking bias of -1.93, underestimating the true effect of 2.0. The Graph-SC (Red) achieves the optimal tradeoff, yielding the lowest Mean Squared Error (MSE) of 0.30 by balancing these objectives. As shown in Figure~\ref{fig:simulation_results} (Bottom), Graph-SC reduces the bias of the sparse estimator by approximately 83\% (from -1.93 to -0.33) while simultaneously maintaining a low variance (0.19) that is significantly better than the baselines.
\begin{figure}[t]
\centering
\begin{minipage}[t]{0.48\linewidth}
\centering
\includegraphics[width=\linewidth]{figures/effect_distribution.png}
\textbf{(a)} Effect distribution
\end{minipage}
\hfill
\begin{minipage}[t]{0.48\linewidth}
\centering
\includegraphics[width=\linewidth]{figures/mse_bias_var.png}
\textbf{(b)} Bias--variance decomposition
\end{minipage}
\caption{\textbf{Bias--Variance Verification.}
\textbf{(a)} Distribution of estimated post-treatment effects across 50 repetitions. The dashed line represents the true average treatment effect on the treated (ATT$=2.0$).
\textbf{(b)} Empirical decomposition of MSE into squared bias and variance. Graph-SC minimizes total error by correcting the heavy bias of Lasso methods while suppressing the variance of graph-only methods.}
\label{fig:simulation_results}
\end{figure}
\subsection{California Proposition 99: Policy Benchmark}
We next evaluate Graph-SC on the canonical California Proposition 99 benchmark by Abadie et al.~\cite{abadie2010synthetic}, which estimates the effect of California's 1988 tobacco-control initiative on per-capita cigarette sales. This experiment serves two purposes. First, Graph-SC should preserve the established qualitative conclusion: cigarette sales in California declined substantially relative to a synthetic counterfactual after Proposition 99. Second, because this benchmark has a well-studied donor pool, it allows us to examine whether graph regularization changes the donor portfolio in a meaningful way rather than merely improving scalar fit.
\noindent\textbf{Setup.}
Following the standard design, we use pre-treatment cigarette sales trajectories before 1988 to construct the synthetic counterfactual. For Graph-SC, the donor graph is built from similarity among pre-treatment donor trajectories. The graph penalty encourages donors with similar pre-treatment evolution to receive coherent weights, while the sparsity penalty prevents the estimator from spreading weight indiscriminately across the donor pool. We compare Graph-SC with classical SC\cite{abadie2003economic}, Synthetic-DiD\cite{arkhangelsky2021synthetic}, AugSC\cite{ben2021augmented}, Generalized-SC\cite{xu2017generalized}, Penalized-SCM\cite{abadie2021penalized}, Robust-SC\cite{amjad2018robust}, Matching+SC\cite{kellogg2021combining}, and two ablations: sparse-only and graph-only regularization.
A useful estimator in this benchmark should satisfy three desiderata simultaneously. First, it should achieve an accurate pre-treatment fit, since poor pre-treatment reconstruction weakens the credibility of the counterfactual. Second, it should preserve the canonical post-treatment discrepancy between California and its synthetic control. Third, it should avoid an overly concentrated donor portfolio, since a highly concentrated solution can be sensitive to idiosyncratic donor choices. We therefore report pre-treatment RMSPE, the post-treatment gap magnitude, the fraction of the SC post-treatment gap retained, the number of active donors, and donor-weight entropy. The retained gap is defined as
$\text{Gap retained}
=
\frac{
|\overline{g}_{post}^{\,m}|
}{
|\overline{g}_{post}^{\,SC}|
}
\times 100\%,
$
where $\overline{g}_{post}^{\,m}$ is the average post-treatment gap for method $m$. The entropy diagnostic is
$
H(w)=-\sum_j w_j \log(w_j+\epsilon),$
where larger values indicate a more diversified donor portfolio.
\textbf{Comparison Results.}
Figure~\ref{fig:prop99_placebo}(a) shows that both SC and Graph-SC closely track California before the 1988 intervention. After treatment, the observed California trajectory drops sharply below both synthetic counterfactuals, reproducing the canonical qualitative finding of Proposition 99. Importantly, Graph-SC does not eliminate the post-treatment discrepancy. Instead, it produces a slightly more conservative but still large post-treatment gap. This behavior is desirable: the graph penalty improves pre-treatment structure and donor diversification without collapsing the substantive post-treatment signal. Figure~\ref{fig:prop99_placebo}(b) reports an in-space placebo test. We reassign the 1988 intervention year to each untreated donor state and compute the corresponding placebo gap using Graph-SC. California's post-treatment gap is extreme relative to the placebo distribution, while its pre-treatment gap remains close to zero. We interpret this result as supporting the benchmark validity of Graph-SC rather than as a new identification claim beyond the standard assumptions of synthetic control.
\begin{figure}[t]
\centering
\begin{minipage}[t]{0.48\linewidth}
\centering
\includegraphics[width=\linewidth]{figures/benchmark_trajectories.png}
\vspace{-0.5em}
\centerline{\small \textbf{(a)} Observed vs. synthetic trajectories}
\end{minipage}
\hfill
\begin{minipage}[t]{0.48\linewidth}
\centering
\includegraphics[width=\linewidth]{figures/benchmark_placebo.png}
\vspace{-0.5em}
\centerline{\small \textbf{(b)} In-space placebo test}
\end{minipage}
\caption{
\textbf{Proposition 99 Benchmark.}
\textbf{(a)} Observed vs. synthetic cigarette sales. Graph-SC closely tracks the pre-treatment trend while reproducing the post-treatment drop reported in \cite{abadie2010synthetic}.
\textbf{(b)} In-space placebo test. The estimated gap for California is extreme compared to the distribution of placebo gaps generated by applying Graph-SC to the 38 untreated donor states, indicating statistical significance.
}
\label{fig:prop99_placebo}
\end{figure}
Table~\ref{tab:prop99_full_benchmark} shows that Graph-SC is the only method satisfying all three diagnostic criteria: very low pre-treatment error, at least $95\%$ retained post-treatment gap relative to SC, and high donor-weight entropy. Compared with classical SC, Graph-SC reduces pre-treatment RMSPE from $1.6564$ to $0.0893$, corresponding to a $94.6\%$ fit gain. At the same time, it preserves $95.4\%$ of the SC post-treatment gap, with $|\overline{g}_{post}|=18.6245$ compared with $19.5136$ for SC. This indicates that Graph-SC substantially improves pre-treatment reconstruction while retaining the main policy-relevant impact.
\begin{table*}[t]
\centering
\begin{threeparttable}
\caption{
\textbf{Proposition 99 benchmark comparison.}
Graph-SC is the only method that simultaneously achieves very low pre-treatment error, preserves at least $95\%$ of the canonical SC post-treatment gap, and produces a high-entropy donor portfolio. Exact-fit methods obtain near-zero pre-RMSPE but substantially attenuate the post-treatment gap, while high-gap alternatives have much worse pre-treatment fit or highly concentrated donor weights.
}
\label{tab:prop99_full_benchmark}
\scriptsize
\setlength{\tabcolsep}{3.5pt}
\begin{tabular}{lrrrrrrrr}
\toprule
\textbf{Method}
& \makecell{\textbf{Pre-RMSPE}\\\textbf{$\downarrow$}}
& \makecell{\textbf{Fit gain}\\\textbf{vs. SC $\uparrow$}}
& \makecell{\textbf{Post-RMSPE}\\\textbf{gap size}}
& \makecell{\textbf{$|\overline{g}_{post}|$}\\\textbf{gap magnitude}}
& \makecell{\textbf{Gap retained}\\\textbf{vs. SC $\uparrow$}}
& \makecell{\textbf{Active}\\\textbf{donors}}
& \makecell{\textbf{Weight}\\\textbf{entropy $\uparrow$}}
& \makecell{\textbf{Criteria}\\\textbf{met}} \\
\midrule
Sparse-only (Ablation)
& \cellcolor{bestgreen}\textbf{0.0058}
& 99.6\%
& 13.0945
& 12.3503
& 63.3\%
& 20
& 2.5791
& 1/3 \\
Graph-only (Ablation)
& \cellcolor{bestgreen}\textbf{0.0663}
& 96.0\%
& 18.4235
& 17.3714
& 89.0\%
& 38
& \cellcolor{bestgreen}\textbf{3.3940}
& 2/3 \\
\rowcolor{methodyellow}
\textbf{Graph-SC (Ours)}
& \textbf{0.0893}
& 94.6\%
& 19.8918
& 18.6245
& 95.4\%
& 38
& \textbf{3.4112}
& \textbf{3/3} \\
\midrule
SC \cite{abadie2003economic}
& 1.6564
& 0.0\%
& 20.6056
& 19.5136
& \cellcolor{bestgreen}\textbf{100.0\%}
& 6
& 1.4752
& 1/3 \\
AugSC \cite{ben2021augmented}
& \cellcolor{bestgreen}\textbf{0.0001}
& 100.0\%
& 13.1322
& 12.3747
& 63.4\%
& 6
& 1.4752
& 1/3 \\
Synthetic-DiD \cite{arkhangelsky2021synthetic}
& 1.6564
& 0.0\%
& 18.8482
& 17.6478
& 90.4\%
& 6
& 1.4752
& 0/3 \\
Generalized-SC \cite{xu2017generalized}
& 1.3718
& 17.2\%
& 7.3630
& 6.4959
& 33.3\%
& 0
& 0.0000
& 0/3 \\
Penalized-SCM \cite{abadie2021penalized}
& 1.6564
& 0.0\%
& 20.6056
& 19.5136
& \cellcolor{bestgreen}\textbf{100.0\%}
& 6
& 1.4752
& 1/3 \\
Robust-SC \cite{amjad2018robust}
& 2.4607
& $-48.6\%$
& 19.0547
& 17.3461
& 88.9\%
& 38
& \cellcolor{bestgreen}\textbf{3.2471}
& 1/3 \\
Matching+SC \cite{kellogg2021combining}
& 3.8165
& $-130.4\%$
& 27.0390
& 25.2279
& \cellcolor{bestgreen}\textbf{129.3\%}
& 4
& 1.2661
& 1/3 \\
\bottomrule
\end{tabular}
\begin{tablenotes}
\footnotesize
\item Note: Green cells indicate that a method satisfies one of three diagnostic criteria: pre-RMSPE $\leq 0.1$, retained post-treatment gap $\geq 95\%$ of SC, or donor-weight entropy $\geq 3.0$. The final column counts how many of these criteria are satisfied. These thresholds are used only as descriptive diagnostics and are not used for training or hyperparameter selection. The Graph-SC row is highlighted in yellow. Post-treatment RMSPE and $|\overline{g}_{post}|$ measure the magnitude of the post-treatment discrepancy rather than supervised prediction error.
\end{tablenotes}
\end{threeparttable}
\end{table*}
The comparison also clarifies why low pre-treatment error alone is not sufficient. AugSC and the sparse-only ablation achieve nearly exact pre-treatment fit, with pre-RMSPE values of $0.0001$ and $0.0058$, respectively. However, both substantially attenuate the post-treatment gap, retaining only $63.4\%$ and $63.3\%$ of the SC gap. These results suggest that exact pre-treatment interpolation can over-adjust the counterfactual and weaken the post-treatment signal. Conversely, Matching+SC produces the largest post-treatment discrepancy, retaining $129.3\%$ of the SC gap, but it has much worse pre-treatment fit with pre-RMSPE $3.8165$ and a concentrated donor portfolio with entropy $1.2661$. Robust-SC has high entropy but also much worse pre-treatment fit than Graph-SC and retains only $88.9\%$ of the SC gap. Thus, the apparent alternatives each fail on at least one diagnostic axis.
\textbf{Role of the graph penalty.}
The ablation results show that the full Graph-SC objective is not simply a cosmetic combination of sparsity and graph regularization. Sparse-only regularization achieves an excellent fit but severely attenuates the post-treatment gap. Graph-only regularization produces a high-entropy donor portfolio but retains only $89.0\%$ of the SC gap. The full Graph-SC estimator balances these effects: it maintains very low pre-treatment RMSPE, preserves the substantive post-treatment gap, and produces the highest donor-weight entropy in the table. This supports the central claim of the paper: graph regularization is most useful when evaluated jointly through fit, signal preservation, and donor-weight stability rather than through a single scalar error metric.
\subsection{Waymo Open Motion Dataset: Observational Counterfactual Contrasts}
We apply Graph-SC to a large-scale safety analysis using the Waymo Open Motion Dataset (WOMD). Prior studies using WOMD have examined scenarios such as human drivers following AVs, AVs following human-driven vehicles, and lane-changing behaviors involving AVs \cite{Chung2025, Zhang2025}. However, a closely related yet distinct question in mixed autonomy traffic remains largely unexplored. Specifically, it is still unclear whether the mere presence of an AV in the surrounding traffic influences the car following behavior of nearby human drivers, particularly given the limited penetration of AVs in real-world traffic.
We analyze Human-Human car-following pairs where an AV becomes visible to the follower ($t \ge 5.0$s, range $<20$m), testing the hypothesis that the presence of AVs acts as a traffic calming stimulus, potentially because human drivers may adopt more conservative spacing to allocate attention to the novel stimulus (the AV) rather than purely forward monitoring. To demonstrate robustness across domains, we apply the same hyperparameters used in the benchmark study ($\lambda_1=0.05, \lambda_g=10.0$). We quantify safety using headway standard deviation (\texttt{headway\_std}), a proxy for flow instability, and minimum Time-to-Collision (\texttt{min\_TTC}), a measure of immediate crash risk. The detailed extraction pipeline of the dataset is provided in Appendix~\ref{app:data_and_setup}.
\begin{table*}[t]
\centering
\begin{threeparttable}
\caption{
\textbf{WOMD multi-baseline robustness comparison.}
Graph-SC is evaluated on 50 AV-exposed car-following events with 300 candidate donors per event.
For each outcome, green cells indicate that a method satisfies one of three diagnostic criteria:
low pre-exposure error, large post-exposure signal relative to the best pre-fit-qualified method, or controlled placebo discrepancy.
For headway variability, Graph-SC is the only method satisfying all three criteria, achieving near-exact pre-exposure fit while producing the largest post-exposure signal among well-fitting methods.
For minimum TTC, Graph-SC and AugSC both satisfy all three criteria; Graph-SC gives a slightly smaller main gap but also the lowest placebo discrepancy and lower cross-unit dispersion.
}
\label{tab:womd_full_benchmark}
\scriptsize
\setlength{\tabcolsep}{3.2pt}
\begin{tabular}{llrrrrrrrr}
\toprule
\textbf{Outcome}
& \textbf{Method}
& \makecell{\textbf{Pre-RMSPE}\textbf{$\downarrow$}}
& \makecell{\textbf{Fit gain}\\\textbf{vs. SC $\uparrow$}}
& \makecell{\textbf{$|\overline{g}_{post}|$}\\\textbf{main gap $\uparrow$}}
& \makecell{\textbf{Gap retained}\\\textbf{vs. best-fit $\uparrow$}}
& \makecell{\textbf{$|\overline{g}_{placebo}|$}\\\textbf{placebo $\downarrow$}}
& \makecell{\textbf{Criteria met}} \\
\midrule
\multirow{8}{*}{\texttt{headway std}}
& SC \cite{abadie2003economic}
& 1.1738
& 0.0\%
& 5.3654
& 82.0\%
& \cellcolor{bestgreen}\textbf{1.2637}
& 1/3 \\
& AugSC \cite{ben2021augmented}
& \cellcolor{bestgreen}\textbf{\textbf{0.0148}}
& 98.7\%
& 6.0753
& 92.9\%
& \cellcolor{bestgreen}\textbf{\textbf{1.2366}}
& 2/3 \\
& Synthetic-DiD \cite{arkhangelsky2021synthetic}
& 1.1738
& 0.0\%
& 5.3654
& 82.0\%
& \cellcolor{bestgreen}\textbf{1.2637}
& 1/3 \\
& Generalized-SC \cite{xu2017generalized}
& 1.6611
& $-41.5\%$
& 12.2329
& \cellcolor{bestgreen}\textbf{187.0\%}
& 2.3937
& 1/3 \\
& Robust-SC \cite{amjad2018robust}
& 2.0516
& $-74.8\%$
& 34.0756
& \cellcolor{bestgreen}\textbf{520.9\%}
& 3.1861
& 1/3 \\
& Penalized-SCM \cite{abadie2021penalized}
& 1.2156
& $-3.6\%$
& 5.5137
& 84.3\%
& \cellcolor{bestgreen}\textbf{1.1103}
& 1/3 \\
& Matching+SC \cite{kellogg2021combining}
& 1.5504
& $-32.1\%$
& 5.0083
& 76.6\%
& 1.3839
& 0/3 \\
\rowcolor{methodyellow}
& \textbf{Graph-SC (Ours)}
& \textbf{0.0236}
& 98.0\%
& 6.5412
& \textbf{100.0\%}
& \textbf{1.3076}
& \textbf{3/3} \\
\midrule
\multirow{8}{*}{\texttt{min TTC}}
& SC \cite{abadie2003economic}
& 0.7203
& 0.0\%
& 1.7767
& 75.1\%
& \cellcolor{bestgreen}\textbf{0.9282}
& 1/3 \\
& AugSC \cite{ben2021augmented}
& \cellcolor{bestgreen}\textbf{\textbf{0.0385}}
& 94.7\%
& 2.3644
& \cellcolor{bestgreen}\textbf{\textbf{100.0\%}}
& \cellcolor{bestgreen}\textbf{1.2018}
& \textbf{3/3} \\
& Synthetic-DiD \cite{arkhangelsky2021synthetic}
& 0.7203
& 0.0\%
& 1.7339
& 73.3\%
& \cellcolor{bestgreen}\textbf{0.9919}
& 1/3 \\
& Generalized-SC \cite{xu2017generalized}
& 1.2394
& $-72.1\%$
& 2.3770
& \cellcolor{bestgreen}\textbf{100.5\%}
& 2.4929
& 1/3 \\
& Robust-SC \cite{amjad2018robust}
& 1.3348
& $-85.3\%$
& 2.7689
& \cellcolor{bestgreen}\textbf{117.1\%}
& 1.3053
& 1/3 \\
& Penalized-SCM \cite{abadie2021penalized}
& 0.7331
& $-1.8\%$
& 1.7563
& 74.3\%
& \cellcolor{bestgreen}\textbf{\textbf{0.8620}}
& 1/3 \\
& Matching+SC \cite{kellogg2021combining}
& 0.9703
& $-34.7\%$
& 2.2626
& \cellcolor{bestgreen}\textbf{95.7\%}
& \cellcolor{bestgreen}\textbf{0.9306}
& 2/3 \\
\rowcolor{methodyellow}
& \textbf{Graph-SC (Ours)}
& \textbf{0.0440}
& 93.9\%
& 2.3171
& \textbf{98.0\%}
& \textbf{1.1786}
& \textbf{3/3} \\
\bottomrule
\end{tabular}
\begin{tablenotes}
\footnotesize
\item Note: Green cells indicate that a method satisfies one of three outcome-specific diagnostic criteria:
pre-RMSPE $\leq 0.1$;
retained main gap $\geq 95\%$ of the largest main gap among pre-fit-qualified methods;
or placebo gap within $110\%$ of the smallest placebo gap among pre-fit-qualified methods.
The final column counts how many criteria are satisfied.
These thresholds are descriptive diagnostics and are not used for training or hyperparameter selection.
The Graph-SC rows are highlighted in yellow.
Large main gaps from poorly fitting methods should not be interpreted as better counterfactual estimates, because they may reflect pre-exposure mismatch rather than post-exposure signal.
\end{tablenotes}
\end{threeparttable}
\end{table*}
Table~\ref{tab:womd_full_benchmark} summarizes the WOMD multi-baseline comparison. For headway, Graph-SC is the only method satisfying all three diagnostic criteria. Compared with standard SC, Graph-SC reduces pre-exposure RMSPE from $1.1738$ to $0.0236$, corresponding to a $98.0\%$ fit gain, while increasing the average absolute post-exposure gap from $5.3654$ to $6.5412$. Compared with AugSC, which has the lowest pre-RMSPE, Graph-SC produces a larger post-exposure signal ($6.5412$ vs. $6.0753$). This result is important because it shows that Graph-SC does not merely improve pre-treatment fit; it selects a graph-coherent counterfactual that preserves a stronger post-exposure contrast.
For min TTC, the comparison is more conservative. Graph-SC and AugSC both satisfy all three criteria. AugSC has slightly lower pre-RMSPE and a slightly larger main gap, whereas Graph-SC has the smallest placebo discrepancy among the pre-fit-qualified methods and slightly lower cross-unit dispersion. Specifically, Graph-SC achieves pre-RMSPE $0.0440$, retains $98.0\%$ of the strongest pre-fit-qualified TTC signal, and has the lowest qualified placebo gap of $1.1786$. This suggests that Graph-SC preserves the immediate-risk dynamics estimated by the strongest exact-fit baseline while avoiding an increase in placebo discrepancy.
\section{Discussion}
This paper studies synthetic control under donor redundancy: settings where many donor portfolios achieve similar pre-intervention fit but yield unstable weights and potentially different post-intervention counterfactuals. Graph-SC addresses this instability by introducing a donor similarity graph learned from pre-intervention data and regularizing the synthetic-control weights through a Laplacian penalty. Across the controlled simulation, the Proposition 99 benchmark, and WOMD observational counterfactual analyses, the main empirical pattern is consistent with the theory: Graph-SC is most useful when pre-treatment fit alone is insufficient to distinguish among many plausible donor mixtures. In this regime, graph regularization improves donor-weight coherence and robustness while preserving the interpretability of weight-based counterfactual construction.
Several practical choices influence Graph-SC performance. The donor graph depends on the pre-intervention feature representation, similarity metric, neighborhood size, and whether edges are weighted or unweighted. Although our ablations in appendix~\ref{app:simulation_graph_ablation} suggest that the method is reasonably stable across several graph choices, graph learning remains an important modeling decision. The diagnostic thresholds used in our tables are intended to summarize tradeoffs among fit, signal preservation, placebo behavior, and donor-weight stability. Promising directions for future work include scalable graph construction via approximate nearest neighbors, richer graph priors such as weighted or multi-view graphs, and joint graph weight learning, extensions to time-varying weights and multi-outcome panels, and uncertainty quantification tailored to graph-regularized counterfactuals.
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