MVBBSC – Bayesian Synthetic Control (Martinez & Vives-i-Bastida 2024)

Estimator:

Bayesian Synthetic Control of Martinez & Vives-i-Bastida (MVBBSC) – mlsynth.MVBBSC

Source:

Martinez, I. and Vives-i-Bastida, J. (2024), “Bayesian and Frequentist Inference for Synthetic Controls,” arXiv:2206.01779 [MVBBSC2024].

Replication type:

cross-validation against the authors’ own reference implementation, the bsynth R package, on the German reunification panel; plus Path A, the German reunification study.

Status:

verified – the NumPyro posterior matches the bsynth (Stan/rstan) posterior to Monte-Carlo error, in both the point estimates and the credible bands.

Validation strategy

Martinez and Vives-i-Bastida (2024) ship their method as the bsynth R package, which fits the model in Stan via rstan. mlsynth’s MVBBSC is a clean-room port of the method – the same generative model (uniform-Dirichlet simplex weights, HalfNormal scale, Gaussian likelihood on the pre-period-standardized series) sampled with NUTS in NumPyro – so bsynth is the ground truth. We fit both on the identical panel and compare the posterior counterfactual, the ATT, and the credible bands.

The reference is the bsynth model bayesianSynth with predictor_match = FALSE (outcome-only, no covariates, no Gaussian-process term), ci_width = 0.95, and four chains of 1000 warm-up + 1000 draws. The mlsynth fit uses the same chain configuration and seed on basedata/german_reunification.csv (West Germany treated, 16 OECD donors, 1960–2003, reunification in 1990).

Cross-validation – point estimates and bands

Both samplers agree to Monte-Carlo error:

Quantity	MVBBSC (NumPyro)	bsynth (rstan)
pre-1990 in-sample RMSE	62.2	62.2
mean post-1990 ATT	\(-2078\)	\(-2075\)
mean 95% band width, pre-1990	300.4	300.9
mean 95% band width, post-1990	1162	1155

Year by year the 95% credible-band widths agree to within a few percent (ratios \(0.95\)–\(1.07\)), the residual being pure Monte-Carlo error between two independent NUTS runs. The band is well calibrated: in-sample coverage of the observed West Germany series is \(96.7\%\) against the nominal \(95\%\), and it has the right shape – tight in the pre-period where the donors pin the fit (\(\sim 300\)), fanning out through the post-period as the counterfactual extrapolates (to \(\sim 2400\) by 2003).

Path A – the reunification result

On its own terms MVBBSC recovers the classical reunification finding. The mean post-1990 effect near \(-2080\) PPP-USD per capita is about a \(7.5\%\) decline in West German per-capita GDP – the magnitude the authors themselves report in the text – with the effect widening after 2000 and a credible band that grows with the horizon.

Reproducing

The durable case benchmarks/cases/mvbbsc_germany.py pins the reproducible headline quantities of a fresh NumPyro fit (the negative effect near the \(7.5\%\) magnitude, a close pre-period fit, an ordered credible band, and converged NUTS). The reference bsynth numbers were produced with the R script under benchmarks/R/mvbbsc_bsynth_ref.R (install pinned via benchmarks/R/install_bsynth.sh); because it needs R and rstan it is not run in CI, but the script is committed so the cross-check is reproducible.