Graph-Regularized Synthetic Control for Stable Counterfactual Estimation under Donor Redundancy

Yanlin Zhang; Sungyong Chung; Alireza Talebpour

cs.LGOct 16, 2026

Graph-Regularized Synthetic Control for Stable Counterfactual Estimation under Donor Redundancy

Authors: Yanlin Zhang, Sungyong Chung, Alireza Talebpour

Organizations: University of Illinois Urbana-Champaign

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.

Figures & tables

Panel of Figure 1 — Figure 1: Bias–Variance Verification. (a) Distribution of estimated post-treatment effects across 50 repetitions. The dashed line represents the true average treatment effect on the treated (ATT $=2.0$ ). (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.

Panel of Figure 2 — Figure 2: Proposition 99 Benchmark. (a) Observed vs. synthetic cigarette sales. Graph-SC closely tracks the pre-treatment trend while reproducing the post-treatment drop reported in Abadie et al. (2010) . (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.

Method	Pre-RMSPE $\downarrow$	Fit gain vs. SC $\uparrow$	Post-RMSPE gap size	$\|\overline{g}_{post}\|$ gap magnitude	Gap retained vs. SC $\uparrow$	Active donors	Weight entropy $\uparrow$	Criteria met
Sparse-only (Ablation)	0.0058	99.6%	13.0945	12.3503	63.3%	20	2.5791	1/3
Graph-only (Ablation)	0.0663	96.0%	18.4235	17.3714	89.0%	38	3.3940	2/3
Graph-SC (Ours)	0.0893	94.6%	19.8918	18.6245	95.4%	38	3.4112	3/3
SC Abadie and Gardeazabal (2003)	1.6564	0.0%	20.6056	19.5136	100.0%	6	1.4752	1/3
AugSC Ben-Michael et al. (2021)	0.0001	100.0%	13.1322	12.3747	63.4%	6	1.4752	1/3
Synthetic-DiD Arkhangelsky et al. (2021)	1.6564	0.0%	18.8482	17.6478	90.4%	6	1.4752	0/3

Table 1: 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.

Outcome	Method	Pre-RMSPE $\downarrow$	Fit gain vs. SC $\uparrow$	$\|\overline{g}_{post}\|$ main gap $\uparrow$	Gap retained vs. best-fit $\uparrow$	$\|\overline{g}_{placebo}\|$ placebo $\downarrow$	Criteria met
headway std	SC Abadie and Gardeazabal (2003)	1.1738	0.0%	5.3654	82.0%	1.2637	1/3
	AugSC Ben-Michael et al. (2021)	0.0148	98.7%	6.0753	92.9%	1.2366	2/3
	Synthetic-DiD Arkhangelsky et al. (2021)	1.1738	0.0%	5.3654	82.0%	1.2637	1/3
	Generalized-SC Xu (2017)	1.6611	$-41.5\%$	12.2329	187.0%	2.3937	1/3
	Robust-SC Amjad et al. (2018)	2.0516	$-74.8\%$	34.0756	520.9%	3.1861	1/3
	Penalized-SCM Abadie and L’hour (2021)	1.2156	$-3.6\%$	5.5137	84.3%	1.1103	1/3

Table 2: 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.

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Estimator / graph setting	ATT MSE $\downarrow$	Bias 2 $\downarrow$	Variance $\downarrow$	Mean Pre-RMSPE $\downarrow$	Mean post-RMSPE vs. noiseless CF $\downarrow$
Sparse-only	0.1074	0.0936	0.0276	0.0057	1.4927
SC	0.1438	0.0449	0.1978	0.9794	1.0407
Graph-only oracle	0.7024	0.6122	0.1803	0.0677	1.6139
Graph-SC oracle	0.5208	0.4747	0.0921	0.0736	1.5328
Graph-SC learned: Euclidean $k=3$	0.5360	0.5204	0.0312	0.0615	1.5816
Graph-SC learned: Euclidean $k=5$	0.5337	0.4952	0.0770	0.0658	1.5374

Appendix

Table 3: Graph-construction ablation in the semi-synthetic study. The table compares sparse-only, standard SC, graph-only, oracle Graph-SC, and learned Graph-SC variants.

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Spectral Truncation in Synthetic Control

Mojtaba Eslami

University of Calgary

Spectral Regularization Causal Effect Estimation Synthetic Control Spectral Methods

Open paper AskPDF arXiv

Synthetic control (SC) matches a treated unit's pre-treatment trajectory to a weighted combination of donor units. We study Spectral SC, which instead matches the treated unit in coordinates defined by the leading temporal singular vectors of the donor panel, and a hybrid estimator that places separately tunable weight on retained and discarded directions, nesting raw-path SC and truncated Spectral SC as endpoints. We prove that the family reduces exactly to raw-path SC at full rank, that exact balance on

K

retained dimensions with

N_0

donors is underdetermined whenever

N_0>K+1

, with an affine solution set of dimension

N_0-K-1

, and that spectral imbalance maps to treatment-effect bias through a finite-sample best-linear-predictor decomposition. We evaluate the estimators across eleven data-generating regimes, using

400

replications per regime and donor-only placebo validation to select regularization and the mixing weight. Truncated Spectral SC has significantly higher RMSE than tuned raw-path SC in every regime, with paired differences equal to

4

11

Monte Carlo standard errors. The hybrid estimator selects raw-path matching in most replications and is statistically indistinguishable from tuned SC in most regimes. The result is highly sensitive to preprocessing. With raw inputs, the performance gap is large; after removing unit and time fixed effects before spectral decomposition, as suggested by the assumptions behind our bound, the gap nearly disappears and placebo validation begins to favor truncation. We interpret these findings diagnostically rather than as evidence that Spectral SC should replace raw-path SC. Basis-estimation noise, balancing underdetermination, and fixed-effects contamination determine when spectral matching can help.

Abstract

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