Bayesian Synthetic Control Methods (BSCM)

Bayesian Synthetic Control Methods (BSCM)#

When to use BSCM – and when not to#

Bayesian Synthetic Control Methods (Kim, Lee and Gupta 2020) are for the case where you want a synthetic control that fits a hard-to-match treated unit, works when the donor pool is large relative to the number of pre-treatment periods, and reports honest finite-sample uncertainty. The canonical synthetic control restricts the donor weights to the unit simplex (non-negative, summing to one), which keeps the counterfactual inside the convex hull of the donors but cannot track a treated unit near or outside that hull, and offers no mechanism for the “large p, small n” problem. BSCM relaxes the simplex and places a Bayesian shrinkage prior on the donor weights, so the counterfactual can extrapolate in a regularised way and every quantity comes with a posterior – credible intervals that fall out of the fit rather than a separate inference step (the classical synthetic control now has its own valid inference options too; see Vanilla Synthetic Control (VanillaSC)). It is a good choice when:

  • You have a single treated unit and a donor pool where a simplex synthetic control leaves a visible pre-treatment gap, and you are willing to let the counterfactual extrapolate a little beyond the donor hull.

  • You have many donors relative to pre-periods (the “large p, small n” regime), where an unpenalised regression would be under-determined. The shrinkage prior identifies the weights without a separate tuning step.

  • You want a credible interval on the effect. Because the sampler returns the full posterior, the ATT and the counterfactual path both carry credible bands directly – valid in finite samples, without asymptotics or a placebo test.

BSCM offers two priors:

  • horseshoe – a global-local continuous shrinkage prior. The default: fast, and it shrinks noise donors hard toward zero while letting genuine signal donors escape via heavy tails.

  • spike_slab – a discrete variable-selection prior. Additionally reports a per-donor inclusion probability \(P(\gamma_j = 1 \mid y)\), so you can read off which donors the model considers signal.

Do not use BSCM when:

The Bayesian SC family#

mlsynth carries three Bayesian synthetic-control estimators; they differ in what they place a prior on, and BSCM is the shrinkage-on-donor-weights member.

Reach for a weighting prior (BSCM, Bayesian Synthetic Control with a Soft Simplex Constraint (BVS-SS)) when you want interpretable donor weights; reach for Bayesian Factor Synthetic Control (BFSC) when a shared factor structure – not a weighted average of donors – is the right model for the untreated outcome.

Notation and the causal estimand#

We use the synthetic-control canon. Suppose we observe \(J + 1\) units over \(T\) periods, with unit \(0\) treated after period \(T_0\) and units \(\{1, \ldots, J\}\) untreated (the donor pool). Let \(Y_{jt}^{N}\) and \(Y_{jt}^{I}\) be the potential outcomes for unit \(j\) at time \(t\) without and with the intervention, and let \(D_{jt} = 1\) if unit \(j\) is under treatment at \(t\) (only \(D_{0t} = 1\) for \(t > T_0\)). The observed outcome is

\[Y_{jt} = Y_{jt}^{N} + \alpha_{jt}\, D_{jt}, \qquad \alpha_{jt} = Y_{jt}^{I} - Y_{jt}^{N}.\]

The object of interest is the treatment effect on the treated unit,

\[\alpha_{0t} = \underbrace{Y_{0t}^{I}}_{\text{observed}} - \underbrace{Y_{0t}^{N}}_{\text{unobserved}}, \qquad t > T_0.\]

Since \(Y_{0t}^{N}\) is never observed after treatment, causal inference reduces to predicting the treated unit’s untreated path from the donor pool – the out-of-sample accuracy of that prediction is what a synthetic control lives or dies by.

The standard synthetic control (Abadie, Diamond and Hainmueller 2010) solves a constrained least-squares problem over the pre-treatment window,

\[\widehat{\boldsymbol\beta} = \arg\min_{\boldsymbol\beta \in \Lambda} \sum_{t=1}^{T_0} \Bigl( Y_{0t} - \beta_0 - \textstyle\sum_{j=1}^{J} \beta_j Y_{jt} \Bigr)^2, \qquad \Lambda = \Bigl\{ \beta_0 = 0,\ \beta_j \ge 0,\ \textstyle\sum_j \beta_j = 1 \Bigr\}.\]

Kim, Lee and Gupta, writing in 2020, identify three limitations of this program. First, the constraints \(\Lambda\) are restrictive: the no-intercept and sum-to-one/non-negativity rules force the synthetic control into the convex hull of the donors and implicitly assume the treated unit is positively correlated with them, biasing the estimator when the treated unit is extremal. Second, the classical estimator’s inference rested on a placebo (permutation) test whose validity relies on a symmetry assumption that is generally violated. Third, there is no explicit mechanism for the “large p, small n” or sparsity problem, where the number of donors approaches or exceeds the number of pre-periods and the unconstrained problem has no unique solution.

The second point should be read in its 2020 context, not as a current verdict. A substantial literature has since given the classical synthetic control a formal inference toolkit – conformal test-inversion intervals (Chernozhukov, Wüthrich and Zhu), prediction intervals (Cattaneo, Feng and Titiunik), subsampling and \(t\)-test approaches, and refined placebo procedures – several of which mlsynth exposes directly on the ordinary synthetic control through the inference option of Vanilla Synthetic Control (VanillaSC). So classical SCM is no longer without valid inference. What BSCM offers is a different and internally coherent route: finite-sample credible intervals that fall out of the posterior as a byproduct of estimation, for the weights, the counterfactual and the ATT at once, without a separate inference procedure.

The Bayesian synthetic-control framework#

BSCM drops the constraint set and writes the pre-treatment relationship as an ordinary linear model with an intercept,

\[Y_{0t} = \beta_0 + \sum_{j=1}^{J} \beta_j\, Y_{jt} + \varepsilon_t, \qquad \varepsilon_t \sim \mathcal{N}(0, \sigma^2), \quad t \le T_0,\]

and regularises the weights \(\boldsymbol\beta\) with a prior rather than a hard constraint. The counterfactual is the fitted line carried past \(T_0\),

\[\widehat{Y}_{0t}^{N} = \beta_0 + \sum_{j=1}^{J} \beta_j\, Y_{jt},\]

so the effective coefficient on donor \(j\) is \(\beta_j\), which may be negative and need not sum to one.

The framework is unifying: the choice of prior on \(\boldsymbol\beta\) recovers a whole family of estimators (Kim, Lee and Gupta 2020, Table 1). An uninformative prior with the simplex constraints reproduces the standard SCM; a normal prior gives ridge regression (the augmented SCM of Ben-Michael, Feller and Rothstein); a Laplace prior gives the lasso (the ArCo of Carvalho, Masini and Medeiros); a scale-mixture-of-normals prior gives the elastic net (the modified SCM of Doudchenko and Imbens). BSCM contributes two priors whose frequentist analogues do not exist:

The horseshoe prior (Carvalho, Polson and Scott 2010) is the global-local shrinkage prior

\[\beta_j \mid \lambda_j \sim \mathcal{N}(0, \lambda_j^2), \qquad \lambda_j \mid \tau \sim \mathcal{C}^+(0, \tau), \qquad \tau \mid \sigma \sim \mathcal{C}^+(0, \sigma), \qquad \sigma \sim \mathcal{C}^+(0, 10),\]

where \(\mathcal{C}^+\) is the half-Cauchy. The global scale \(\tau\) shrinks every weight toward zero; the local scales \(\lambda_j\) have heavy (half-Cauchy) tails that let a genuine signal escape that shrinkage. The resulting marginal prior on \(\beta_j\) has a tall spike at zero and flat tails – it separates noise donors (shrunk hard) from signal donors (left alone), which is exactly the behaviour a sparse synthetic control needs.

The spike-and-slab prior (George and McCulloch 1993; Mitchell and Beauchamp 1988) is the discrete mixture

\[\beta_j \sim \gamma_j\, \mathcal{N}(0, \tau_j^2) + (1 - \gamma_j)\, \mathcal{N}(0, c^2), \qquad \gamma_j \sim \text{Uniform}(0, 1), \qquad \tau_j^2 \sim \text{Inv-Gamma}\!\left(\tfrac12, \tfrac12\right),\]

with a small fixed spike variance \(c^2\) (the paper uses \(c^2 = 0.001\)). The spike concentrates a donor at zero; the slab is a diffuse prior for donors that carry signal; the inclusion indicator \(\gamma_j\) has posterior mean equal to the probability that donor \(j\) is a signal. The spike-and-slab is the reference sparse-estimation prior; the horseshoe is a continuous relaxation of it that is typically faster to sample and, when no small set of donors clearly dominates, shrinks more gracefully.

Three advantages follow from being Bayesian. First, finite-sample inference comes for free: because MCMC returns samples from the posterior, credible intervals for the weights, the counterfactual and the ATT are all available directly from one fit, with no asymptotics and no separate inference procedure, and this holds even when \(n\) is small. This is not a claim that the frequentist synthetic control cannot do inference – as noted above, it now can, several ways – but a coherent alternative in which the interval is a byproduct of estimation rather than a bolt-on, and which sidesteps the instability of bootstrapping a penalised regression whose coefficients sit near zero. Second, the shrinkage prior is a built-in mechanism for the “large p, small n” and sparsity problems – the third limitation – so BSCM remains well posed when \(J \gtrsim T_0\). Third, the penalty (the shrinkage scales) and the coefficients are estimated jointly in one posterior, avoiding the “double-shrinkage” bias of the frequentist elastic net. In the paper’s simulations, across data-generating processes that are sparse, dense, low-\(n\), or that violate the SCM constraints, the horseshoe and spike-and-slab almost always match or beat the standard SCM and the other Bayesian regularised regressions on leave-one-out predictive accuracy.

Assumptions#

  1. Single treated unit, balanced panel. One unit is treated after \(T_0\); every unit is observed at every period.

    Remark. The setup boundary (mlsynth.utils.bscm_helpers.setup) enforces both, translating a violation to MlsynthDataError.

  2. Enough pre-treatment periods to identify the weights. Formally BSCM needs at least two pre-periods, but the practical requirement is stronger: the shrinkage prior can only pin down a stable sparse fit if \(T_0\) is large enough. What matters is the absolute number of pre-periods, not the donor-to-period ratio.

    Remark. When \(T_0\) is small relative to \(J\), the model can interpolate the pre-period – the in-sample fit looks near-perfect but does not generalise. Judge the fit by held-out or leave-one-out prediction and by the width of the credible band, never by the in-sample pre-RMSE. See “Inference and diagnostics” below.

  3. Regularised extrapolation is acceptable. The weights can leave the simplex, so the counterfactual can extrapolate; the shrinkage prior bounds how much.

    Remark. If extrapolation is unacceptable (you need a convex, interpretable weighting), BSCM is the wrong tool – use a simplex SCM.

Inference and diagnostics#

BSCM returns a full posterior. The ATT is the posterior mean of the mean post-treatment gap, and res.att_ci is its credible interval at level \(1 - \text{ci\_alpha}\). The counterfactual path carries pointwise credible bands (res.inference_detail.counterfactual_lower / counterfactual_upper). The posterior-mean donor weights res.donor_weights may be negative; for the spike_slab prior, res.inclusion_probs gives the per-donor posterior probability of being a signal. Multiple chains (chains) are sampled and pooled, enabling convergence checks. The sampler is seeded (seed), so a given call is reproducible.

Overfitting is real, and the in-sample pre-fit does not diagnose it. Relaxing the simplex removes a strong regularizer, so with a large donor pool and few pre-periods the unconstrained regression can drive the in-sample pre-RMSE to almost zero while generalising poorly – a beautiful pre-fit that means nothing. The honest diagnostics are the ones the paper uses: leave-one-out cross-validation, or a held-out pre-period test (hold out the last few pre-treatment periods, where the true effect is zero, and check the prediction there). The credible band is a second, automatic guard – when many weightings fit the pre-period equally well the posterior widens, so a wide band on the ATT is the model telling you the pre-fit is less informative than it looks. Two worked cases make the point. On the China anti-corruption watch panel (87 donors, 35 pre-periods, genuinely \(p > n\)) the horseshoe pre-RMSE equals the regularised simplex fit and the held-out error is close to the in-sample error: the shrinkage regularises rather than interpolates. On the Basque panel (16 donors, 15 pre-periods, \(p \approx n\)) the same estimator interpolates to a near-zero pre-RMSE whose held-out error is dozens of times larger: a cautionary case, not a validation. The difference is the number of pre-periods, not the donor-to-period ratio.

Example#

import pandas as pd
from mlsynth import BSCM

# China anti-corruption watch-demand panel: 1 treated series ("watches"),
# 87 donor series, treatment at month index 35 (2013-01).
df = pd.read_csv("basedata/china_watches_long.csv")

res = BSCM({
    "df": df, "outcome": "y", "treat": "treat",
    "unitid": "unit", "time": "time",
    "prior": "horseshoe", "seed": 2019,
    "display_graphs": False,
}).fit()

lo, hi = res.att_ci
print(f"pre-period RMSE : {res.pre_rmse:.3f}")
print(f"ATT            : {res.att:+.3f}  [{lo:+.3f}, {hi:+.3f}]")
top = max(res.donor_weights, key=lambda k: abs(res.donor_weights[k]))
print(f"dominant donor : {top} = {res.donor_weights[top]:+.3f}")

With 87 donors and 35 pre-periods this is the “large p, small n” regime BSCM is built for. The horseshoe shrinks all but a handful of donors to near-zero, concentrating on a dominant donor (C60), fits the pre-period to about the same RMSE as a simplex SCM (no interpolation), and returns an ATT whose 95% credible interval crosses zero – no statistically credible effect of the 2013 anti-corruption campaign on this residualised watch-demand series. Switch "prior": "spike_slab" to additionally read off res.inclusion_probs.

Verification#

BSCM’s horseshoe is cross-validated three ways on the China anti-corruption watch panel (china_watches_long.csv), the “large p, small n” benchmark for the forward-selection panel-data approach. Against the authors’ reference Stan horseshoe (clarencejlee/bscm, sampled with rstan) the pure-numpy Gibbs matches essentially exactly – the ATT agrees to four decimals (\(-0.0221\)), the pre-RMSE and dominant donor C60 agree, and the credible bands coincide. Against the forward-selected panel-data approach (mlsynth.PDA with method="fs", the dataset’s original benchmark) it selects the same dominant donor and returns the same near-null ATT. Against a simplex SCM the horseshoe pre-fit equals the regularised simplex fit, confirming that the shrinkage regularises rather than interpolates in this regime. See the replication page BSCM — Bayesian Synthetic Control Methods (Kim, Lee & Gupta 2020) and the durable case benchmarks/cases/bscm_china_watches.py. The sampler, setup, inference, plotter and result contract are unit-tested (mlsynth/tests/test_bscm.py, full coverage).

Core API#

Bayesian Synthetic Control Methods (BSCM).

Implements:

Kim, S., Lee, C., & Gupta, S. (2020). “Bayesian Synthetic Control Methods.” Journal of Marketing Research 57(5):831-852.

BSCM regresses the treated unit on the donor pool with no simplex constraint (weights may be negative and need not sum to one) and regularises the weights with a Bayesian global-local shrinkage prior – a horseshoe (continuous) or a spike_slab (discrete variable selection). Posterior samples are drawn by a pure-numpy Gibbs sampler, yielding a counterfactual with credible bands and an ATT credible interval with no MCMC-engine dependency.

See mlsynth.utils.bscm_helpers for the algorithmic pieces.

class mlsynth.estimators.bscm.BSCM(config: BSCMConfig | dict)#

Bases: object

Bayesian Synthetic Control Methods (Kim, Lee & Gupta 2020).

Parameters:

config (BSCMConfig or dict) – Configuration object. See mlsynth.config_models.BSCMConfig.

Returns:

BSCMResults – Posterior counterfactual with credible bands, the ATT posterior mean + credible interval, posterior-mean donor weights (which may be negative), and – for spike_slab – per-donor inclusion probabilities.

fit() → BSCMResults#

Run the BSCM Gibbs sampler and assemble results.

Configuration#

class mlsynth.config_models.BSCMConfig(*, df: ~pandas.DataFrame, outcome: str, treat: str, unitid: str, time: str, display_graphs: bool = False, save: bool | str = False, counterfactual_color: ~typing.List[str] = <factory>, treated_color: str = 'black', plot: ~mlsynth.config_models.PlotConfig = <factory>, prior: ~typing.Literal['horseshoe', 'spike_slab'] = 'horseshoe', n_iter: ~typing.Annotated[int, ~annotated_types.Ge(ge=10)] = 2000, burn_in: ~typing.Annotated[int, ~annotated_types.Ge(ge=0)] = 1000, chains: ~typing.Annotated[int, ~annotated_types.Ge(ge=1)] = 4, spike_scale: ~typing.Annotated[float, ~annotated_types.Gt(gt=0)] = 0.03162277660168379, ci_alpha: ~typing.Annotated[float, ~annotated_types.Gt(gt=0.0), ~annotated_types.Lt(lt=1.0)] = 0.05, verbose: bool = False, seed: int | None = None)#

Configuration for the Bayesian Synthetic Control Methods (BSCM).

Implements:

Kim, S., Lee, C., & Gupta, S. (2020). “Bayesian Synthetic Control Methods.” Journal of Marketing Research 57(5):831-852.

BSCM regresses the treated unit on the donor pool with no simplex constraint (weights may be negative and need not sum to one), regularising the weights with a Bayesian global-local shrinkage prior. Two priors are offered: the horseshoe (global-local continuous shrinkage) and the spike_slab (discrete variable selection). Posterior samples are drawn by a pure-numpy Gibbs sampler, giving a counterfactual with credible bands and an ATT credible interval without any MCMC-engine dependency.

Parameters:

  • prior ({“horseshoe”, “spike_slab”}) – Shrinkage prior on the donor weights. horseshoe is the paper’s default (faster, continuous); spike_slab additionally reports per-donor inclusion probabilities.

  • n_iter (int) – Total Gibbs iterations per chain (including burn-in).

  • burn_in (int) – Warm-up iterations discarded per chain before reporting.

  • chains (int) – Number of independent chains, sampled in one vectorised loop and pooled. Multiple chains enable convergence diagnostics.

  • spike_scale (float) – Standard deviation of the spike component for spike_slab (the paper fixes the spike variance to 0.001, i.e. scale ~0.0316). Ignored for the horseshoe.

  • ci_alpha (float) – Two-sided significance level for credible intervals (0.05 gives 95% bands).

  • display_graphs (bool) – Display the observed-vs-counterfactual plot after the fit.

  • verbose (bool) – Reserved for progress reporting.

  • seed (int, optional) – Seed for the numpy.random.Generator used inside the sampler.

burn_in: int#
chains: int#
ci_alpha: float#
display_graphs: bool#
model_config: ClassVar[ConfigDict] = {'arbitrary_types_allowed': True, 'extra': 'forbid'}#

Configuration for the model, should be a dictionary conforming to [ConfigDict][pydantic.config.ConfigDict].

n_iter: int#
prior: Literal['horseshoe', 'spike_slab']#
seed: int | None#
spike_scale: float#
verbose: bool#

Result Containers#

BSCM.fit() returns a BSCMResults – an EffectResult whose standardized sub-models carry the ATT, counterfactual, gap and pre-RMSE, with the posterior draws, per-draw ATT samples, counterfactual credible bands and (for spike_slab) inclusion probabilities on the typed fields. The prepared NumPy panel is exposed as a BSCMInputs.