Bayesian Synthetic Control Method

The classical SCM finds a single optimal weight vector by minimising pre-treatment prediction error subject to convexity constraints. The Bayesian version places a prior over the weights, updates it with pre-treatment data to obtain a posterior, and then propagates that uncertainty into the counterfactual prediction. The result is a distribution of plausible counterfactuals rather than a single trajectory, enabling credible intervals rather than relying solely on permutation tests.

CausalImpact is an open-source R package by Brodersen et al. (2015) that operationalises Bayesian causal inference for time series. It uses a Bayesian structural time-series model — which can incorporate donor-series predictors — to construct a probabilistic counterfactual. It is the most widely used Bayesian SCM implementation and produces posterior credible bands automatically.

There is no hard minimum, but shorter pre-periods produce weaker identification of the weights and wider, less informative posteriors. As a rough guide, at least 20-30 pre-intervention periods provide enough signal to obtain a useful posterior. With fewer periods the credible intervals are likely to be too wide for practical inference.

Common choices are a Dirichlet distribution (naturally on the simplex) or a spike-and-slab prior that induces sparsity. The regularised horseshoe or Laplace priors are also used when sparse donor selection is desired. In all cases, conduct a prior-sensitivity analysis by repeating the analysis with alternative priors to check whether conclusions are robust.

The Bayesian SCM is designed for the case of a single (or very few) treated units. When many units are treated, panel methods such as difference-in-differences or two-way fixed effects models are more natural and statistically efficient. With multiple treated units you can run separate Bayesian SCMs per unit and combine them, but purpose-built multi-unit estimators are usually preferable.