sar — the Bayesian spatial-autoregressive spillover model
Sakaguchi & Tagawa (2026): identification and two-step Bayesian inference
sar is scspill’s first model, and today its only one — the model catalogue lists what else is planned. It implements Sakaguchi & Tagawa’s Identification and Bayesian Inference for Synthetic Control Methods with Spillover Effects. This page states the model, the identification result, and the samplers, and documents precisely where this implementation deliberately departs from the authors’ R replication package.
Select it with method="sar" in SCSPILLConfig. It is the default, so existing code needs no change.
Setup
Units \(i = 0, 1, \dots, N\) are observed over periods \(t = 1, \dots, T\). Unit \(0\) receives a treatment from period \(T_0 + 1\) onward; units \(1, \dots, N\) are the donors. Potential outcomes are indexed by the entire treatment vector — SUTVA is not assumed — so the donors may be affected by unit \(0\)’s treatment. Two estimands follow:
- the treatment effect on the treated, \(\xi_{0t} = Y_{0t}(1, 0, \dots, 0) - Y_{0t}(0, \dots, 0)\), and
- the spillover effect on donor \(i\), \(\xi_{it} = Y_{it}(1, 0, \dots, 0) - Y_{it}(0, \dots, 0)\).
The synthetic-control assumption is perfect fit with unconstrained weights: there exists \(\alpha \in \mathbb{R}^N\) with \(Y_{0t}(\mathbf{0}) = \sum_{i \ge 1} \alpha_i\, Y_{it}(\mathbf{0})\) for all \(t\). No simplex constraint — weights may be negative or exceed one.
Why classical SCM breaks. The observed gap decomposes as \[ Y_{0t} - \sum_i \alpha_i Y_{it} = \xi_{0t} + \underbrace{\textstyle\sum_i \alpha_i \bigl(Y_{it}(\mathbf 0) - Y_{it}(1, \mathbf 0)\bigr)}_{\text{spillover bias}}, \] so whenever the donors absorb part of the treatment, the classical estimate mixes the effect with the contamination.
The spillover model
Spillovers travel through a spatial-autoregressive (SAR) structure on the donor outcomes: \[ \mathbf Y^c_t \;=\; \rho\,\bigl(\mathbf w\, Y_{0t} + \mathbf W\, \mathbf Y^c_t\bigr) \;+\; \mathbf X_t \boldsymbol\beta \;+\; \mathbf u_t , \] where \(\mathbf w \in \mathbb R^N\) measures each donor’s exposure to the treated unit (contiguity, trade, any a-priori channel), \(\mathbf W \in \mathbb R^{N \times N}\) links donors to each other, and one scalar \(\rho\) scales the whole channel. \(\mathbf X_t\) are optional covariates and the errors follow a latent AR(1) factor model, \(\mathbf u_t = \boldsymbol\eta\, \boldsymbol\gamma_t + \mathbf e_t\). At \(\rho = 0\) the model — and the estimator — collapses exactly to the Bayesian horseshoe synthetic control.
Identification
Let \(\mathbf A = \mathbf W + \mathbf w\, \boldsymbol\alpha^\top\) and require \(\mathbf I_N - \rho \mathbf A\) to be invertible. Substituting the perfect-fit assumption into the SAR system yields the donors’ no-treatment counterfactual in closed form: \[ \mathbf Y^c_t(\mathbf 0) = (\mathbf I_N - \rho \mathbf A)^{-1} \bigl[(\mathbf I_N - \rho \mathbf W)\, \mathbf Y^c_t - \rho\, \mathbf w\, Y_{0t}\bigr], \] and with it both estimands: \[ \xi_{0t} = Y_{0t} - \boldsymbol\alpha^\top \mathbf Y^c_t(\mathbf 0), \qquad \boldsymbol\xi^c_t = \mathbf Y^c_t - \mathbf Y^c_t(\mathbf 0). \] Only \((\boldsymbol\alpha, \rho, \mathbf w, \mathbf W)\) and the observed outcomes enter — the covariate coefficients, factors, and error variances cancel out of the effects.
Two-step Bayesian inference
The joint pre-treatment likelihood contains the product \(\rho\, \mathbf w \boldsymbol\alpha^\top\), which is weakly identified and mixes badly, so the paper factorizes it and samples sequentially (a cut posterior):
Step 1 — synthetic weights. A Bayesian regression of \(Y_{0t}\) on \(\mathbf Y^c_t\) over the pre-period with the horseshoe prior \[ \alpha_i \mid \lambda_i \sim \mathcal N(0, \lambda_i^2), \quad \lambda_i \mid \tau \sim \mathcal C^+(0, \tau), \quad \tau \sim \mathcal C^+(0, \sigma_1), \quad \sigma_1 \sim \mathcal C^+(0, 10), \] sampled with the Makalic–Schmidt (2015) inverse-gamma auxiliary representation, so every full conditional is closed form. The outcomes are scaled by their standard deviations only (no centering), and the draws are back-transformed.
Step 2 — the SAR block. With \(\boldsymbol\alpha\) fixed at its posterior mean \(\hat{\boldsymbol\alpha}\), a Gibbs sampler draws the latent AR(1) factors (forward-filter backward-sample), the covariate coefficients \(\boldsymbol\beta\) (horseshoe), the error variance \(\sigma^2\) (inverse-gamma), and \(\rho\) by random-walk Metropolis. The \(\rho\) step’s Jacobian \(\log\lvert \mathbf I - \rho \mathbf A\rvert\) is evaluated in \(O(N)\) per proposal from the pre-computed complex eigenvalues of \(\mathbf A\), and \(\rho\) lives on the spectral support \(|\rho| < 0.95 / \max(1, \max_i |\lambda_i(\mathbf W)|)\).
Effects. Each retained draw pair \((\boldsymbol\alpha^{(m)}, \rho^{(m)})\) is plugged into the identification formulas; posterior means and equal-tailed quantiles give the point estimates and credible bands. The per-draw inverse uses a one-time eigendecomposition of \(\mathbf W\) plus the Sherman–Morrison identity for the rank-one term \(\rho\, \mathbf w \hat{\boldsymbol\alpha}^\top\), so the whole posterior sweep costs \(O(M N^2 T)\).
Where sar departs from the R replication package
These differences belong to this model’s port, not to scspill as a whole. sar defaults to the paper-correct behavior and documents every difference; several came out of validating the port. Each has either an escape hatch or a benchmark demonstrating the consequence.
| # | R replication package | scspill default | Escape hatch |
|---|---|---|---|
| 1 | The covariate array reaches the C++ sampler through mismatched memory layouts, so \(\mathbf X_t\) entered the published fits as scrambled noise | covariates carried as a proper \((T, N, K)\) array end-to-end | drop covariates to reproduce the R fits’ effective specification |
| 2 | \(\boldsymbol\beta\) sampled under a flat-plus-ridge conditional despite the paper’s horseshoe | the paper’s horseshoe on \(\boldsymbol\beta\) | beta_prior="ridge" |
| 3 | ATT credible intervals vary only \(\rho\), holding \(\boldsymbol\alpha\) at its posterior mean | paired \((\boldsymbol\alpha^{(m)}, \rho^{(m)})\) draws (the paper’s stated procedure) | propagate_alpha=False |
| 4 | fixed Metropolis step for \(\rho\) (empirical ESS as low as 3–44) | Robbins–Monro adaptation during burn-in toward 44% acceptance, then frozen | adapt_rho=False |
| 5 | the factor block’s \(\omega_k\) conditional treats \(\omega_k\) as a variance multiplier while its \(\boldsymbol\eta\) / \(\sigma_\eta^2\) conditionals treat it as a precision — mutually inconsistent conditionals; its hyperprior also encodes \(\omega_k \sim \mathcal C^+(0, 1)\) where the paper states \(\mathcal C^+(0, 10)\) | the paper’s parametrization \(\eta_{ik} \sim \mathcal N(0, \sigma^2_\eta \omega_k)\) with \(\omega_k \sim \mathcal C^+(0, 10)\) throughout | — (the R combination is not a valid Gibbs sampler for any posterior) |
| 6 | the FFBS initializes \(\gamma_1\) from the stationary AR(1) variance while the \(\phi_\gamma\) / \(\sigma^2_\gamma\) conditionals assume \(\gamma_1 \sim \mathcal N(0, \sigma^2_\gamma)\) | the coherent \(\gamma_0 = 0\) initialization | — |
Items 5–6 were caught by the Geweke joint distribution test — with the fixes, every sampler block passes it. Item 1 has a striking consequence: fitted without covariates, scspill reproduces the R California posterior almost exactly (\(\hat\rho = 0.187\) vs. R’s \(0.185\)); with the retail-price covariate entering correctly, \(\hat\rho\) moves to roughly \(0.35\). The benchmark suite (benchmarks/REPORT.md) gates both claims.
References
- Sakaguchi, S., & Tagawa, H. (2026). Identification and Bayesian Inference for Synthetic Control Methods with Spillover Effects. The Econometrics Journal. https://doi.org/10.1093/ectj/utag006
- Abadie, A., Diamond, A., & Hainmueller, J. (2010). Synthetic control methods for comparative case studies. JASA 105(490).
- Carvalho, C., Polson, N., & Scott, J. (2010). The horseshoe estimator for sparse signals. Biometrika 97(2).
- Makalic, E., & Schmidt, D. (2015). A simple sampler for the horseshoe estimator. IEEE Signal Processing Letters 23(1).
- Kim, S., Lee, C., & Gupta, S. (2020). Bayesian synthetic control methods. Journal of Marketing Research 57(5).
- Pang, X., Liu, L., & Xu, Y. (2022). A Bayesian alternative to synthetic control for comparative case studies. Political Analysis 30(2).
- Geweke, J. (2004). Getting it right: joint distribution tests of posterior simulators. JASA 99(467).
- LeSage, J., & Pace, R. K. (2009). Introduction to Spatial Econometrics.