Network-Aware Synthetic Control (NASC)
The synthetic control method recovers credible counterfactuals under SUTVA, an assumption rarely plausible in spatial settings such as transport investments, place-based policies, or regional taxation. Effects propagate through trade, commuting, and competition networks to precisely those donor units from which the counterfactual is constructed, so that a convex combination of contaminated donors inherits part of the policy’s impact and yields a biased treatment effect estimate.
The Network-Aware Synthetic Control (NASC) estimator embeds the donor pool in a spatial autoregressive outcome model on a known network and addresses interference through two complementary channels. A contamination penalty discounts each donor in proportion to its network exposure, sidestepping the necessity of restricting the donor pool ex ante, while a bias correction accounts for the remaining contamination that the selected donors inherit. The estimator requires neither a clean donor nor an exposure pattern specified beyond the network itself, and its bias vanishes asymptotically under perfect pre-treatment balance.
Exposure along the network
Interference is assumed to travel across a known undirected graph with row-standardized weight matrix \(W\). The common factor model underlying the conventional synthetic control (see Abadie et al. 2010) is embedded in a spatial Durbin model,
\[y_t(0) = (I_{J+1} - \rho W)^{-1}\big(Z\beta_t + WZ\theta_t + M\lambda_t + \nu_t\iota_{J+1} + \varepsilon_t\big),\]where the autoregressive parameter \(\rho \in (-1,1)\) proxies the strength of spillovers. Partitioning \(W\) into the links \(w_{J1}\) running from the treated unit to the donors and the donor-to-donor block \(W_J\) yields the contamination vector
\[s \;\equiv\; \rho\,(I_J - \rho W_J)^{-1} w_{J1}, \qquad \delta_t \;=\; s\,\tau_{1t},\]so that donor \(i\) inherits the share \(s_i\) of the treatment effect. Expanding the inverse as a Neumann series \(\sum_{k\ge 0}(\rho W_J)^k\) decomposes \(s_i\) into successive orders of network exposure, accumulating over paths of increasing length connecting donor \(i\) to the treated unit, each discounted by a power of \(\rho\), so exposure decays with network distance.
This makes the bias of the conventional estimator explicit. For a weight vector \(\gamma^{sc}\) on the simplex, the synthetic control estimate decomposes as
\[\tau^{sc}_{1t} \;=\; \kappa^{sc}_{1t} \;+\; \big(1 - \gamma^{sc\prime} s\big)\,\tau_{1t},\]where \(\kappa^{sc}_{1t}\) is the usual post-treatment prediction error. Interference enters as a multiplicative bias through the scalar \(\gamma^{sc\prime}s\), additively separable from \(\kappa^{sc}_{1t}\) and, unlike it, not vanishing as the pre-treatment period grows. For positive spillovers the bias is strictly attenuating; for negative spillovers its sign depends on the network and the realized weights.
Regularization and correction
NASC accounts for the interference bias via two channels. First, the synthetic control objective is augmented with a penalty on the weighted absolute contamination, discounting each donor in proportion to its exposure:
\[\hat\gamma^{\,nasc}(\lambda) \;=\; \arg\min_{\gamma \in \Delta^{J-1}}\; \lVert X_1 - X_0\gamma \rVert_V^2 \;+\; \lambda\,\gamma'\lvert s\rvert .\]As \(\lambda \to 0\) the objective collapses to the conventional synthetic control; as \(\lambda\) grows, mass shifts from heavily exposed donors towards weakly contaminated ones. Donor selection is thereby endogenized rather than restricted ex ante. A contaminated donor still contributes whenever its value for the pre-treatment fit outweighs the contamination it introduces. Any \(\lambda > 0\) additionally resolves the familiar non-uniqueness of synthetic weights by selecting the least contaminated among fit-equivalent solutions, and \(\lambda\) itself is chosen by cross-validation on the pre-treatment period.
Second, a bias correction rescales the contamination the selected donors still inherit. With \(\tau^{cr}_{1t} = y_{1t} - \gamma^{nasc\prime}y^{d}_{t}\) denoting the penalized synthetic control estimate, the NASC estimators of the direct and indirect effects are
\[\hat\tau^{\,nasc}_{1t} \;=\; \frac{\tau^{cr}_{1t}}{1-\gamma^{nasc\prime}s}, \qquad \hat\delta^{\,nasc}_{t} \;=\; \frac{s}{1-\gamma^{nasc\prime}s}\,\tau^{cr}_{1t}.\]Because regularization has already shrunk \(\gamma'\lvert s\rvert\), the correction factor sits closer to unity and amplifies the remaining prediction error less than correcting unpenalized weights would. The second expression delivers the implied spillover onto every untreated unit as a by-product. Under perfect pre-treatment balance the estimation error reduces to \((1-\gamma^{nasc\prime}s)^{-1}\kappa^{nasc}_{1t}\), which is bounded by a quantity that vanishes as the pre-treatment period expands.
Modular Bayesian inference
Since \(s\) is a function of the estimated \(\rho\), the correction is itself a random variable. Inference places an exponentially tilted Dirichlet prior on the weights, which encodes the contamination penalty directly on the simplex,
\[\gamma \mid \lambda, \rho \;\sim\; \mathrm{ETDir}\big(\alpha, \lambda\lvert s\rvert\big) \;\propto\; \mathrm{Dir}(\gamma;\alpha)\,\exp\!\big(-\lambda\,\gamma'\lvert s\rvert\big), \qquad \gamma \in \Delta^{J-1},\]and whose maximum-a-posteriori estimate under a concentration parameter \(\alpha = 1\) reproduces the frequentist NASC weights exactly. To preserve uncertainty quantification estimation then proceeds along a cut posterior,
\[p_{\mathrm{cut}}(\zeta \mid Y, Z, W) \;=\; p(\zeta_2 \mid X_1, W, \zeta_1)\;p(\zeta_1 \mid Y, Z, W),\]with \(\zeta_1=\{\rho,\beta,\sigma_1\}\) estimated on pre-treatment data in an initial module and the weights \(\zeta_2=\{\gamma,\lambda,\sigma_2\}\) in a second module conditional on the first. Information flows strictly forward as uncertainty about \(\rho\) propagates into the weights and the bias correction, while the synthetic control matching cannot feed back and misspecify the underlying network dependence. Monte Carlo simulations across regular and small-world topologies show that the proposed estimator outperforms the conventional and the Bayesian synthetic control across various interference structures.
Hands-On
The hands-on example below calculates the NASC estimator on a four-unit network with a single treated unit. In this case the weight simplex \(\Delta^{2}\) reduces to a two-dimensional triangle. Toggle edges to change the network, and the contamination vector, the weights, the counterfactual and the tilted prior all update together.
Donor untreated potential outcomes solve the spatial autoregression
\[y^{d}_{t}(0) = \rho\,w_{J1}\,y_{1t}(0) + \rho\,W_{J}\,y^{d}_{t}(0) + \mu + Z_{t}'\beta + \varepsilon_{t},\]with \(\beta = (1, 0.5)'\), unit effects \(\mu_i \sim N(0,1)\), standard normal covariates and \(\varepsilon_{it} \sim N(0, 0.5^2)\). The treated unit’s own untreated path is the convex combination \(\gamma^{*\prime}(\mu + Z_t'\beta)\) with \(\gamma^{*} = (0.2, 0.5, 0.3)'\), which places the largest true weight on the donor adjacent to the treated unit. Donors that contribute substantially to the pre-treatment fit are also the most contaminated. Treatment occurs at \(t = 25\) with \(\tau_t \sim N(5,1)\), and donor \(i\) inherits \(s_i\tau_t\).
The method is implemented in the R package nasc, which extends the estimator to a spatial Durbin random-effects panel specification. NASC is open-source software. Visit the repository for installation instructions, examples, and the latest development version.