Synthetic Control Method

Unlike traditional difference-in-differences which relies on a single control group, SCM creates a synthetic control—a weighted average of multiple untreated units. The weights are optimized to minimize the pre-treatment root mean squared prediction error (RMSPE) between the treated unit and the synthetic version.

• Weights are non-negative and sum to 1, enforcing a convex hull constraint
• Predictor variables include pre-treatment outcomes and covariates
• The method formalizes the case study logic by making the comparison unit selection explicit and algorithmic

Since SCM typically involves a single treated unit, standard large-sample inference is unavailable. Instead, in-space placebo tests are used: the treatment is fictitiously assigned to each untreated unit in the donor pool, and synthetic controls are constructed for each.

• The treatment effect is compared against the distribution of placebo effects
• A p-value is derived as the proportion of placebos with effects as extreme as the treated unit
• This non-parametric approach avoids distributional assumptions about the error term

The weight vector is chosen through a nested optimization procedure. An inner loop minimizes the discrepancy in predictor variables, while an outer loop minimizes the pre-treatment outcome path divergence.

• Predictor importance is determined by a diagonal matrix V, which can be data-driven or user-specified
• Cross-validation within the pre-treatment period selects V to minimize out-of-sample prediction error
• The optimization is constrained to ensure the synthetic control lies within the convex hull of donor units

SCM provides full transparency into the counterfactual composition. Each donor unit's contribution is explicitly quantified, allowing researchers to assess the face validity of the synthetic control.

• The weight vector reveals which units drive the comparison
• Sparse solutions with few non-zero weights are preferred for interpretability
• This contrasts with black-box methods like causal forests, where the counterfactual logic is opaque
• Stakeholders can evaluate whether the synthetic unit makes substantive sense

The original SCM framework handles a single treated unit, but extensions accommodate multiple treated units with staggered adoption timing.

• Generalized Synthetic Control (GSC) combines interactive fixed effects with SCM logic
• Augmented SCM uses ridge regression to handle cases where the treated unit lies outside the donor pool's convex hull
• Matrix completion methods treat the untreated potential outcomes as a low-rank matrix estimation problem
• These extensions maintain the core SCM philosophy while relaxing restrictive assumptions

A rigorous SCM analysis includes multiple sensitivity checks to validate the estimated treatment effect.

• Leave-one-out analysis: iteratively removes each donor unit with positive weight to test stability
• Backdating: shifts the treatment date to an earlier placebo period to verify no effect appears when none should exist
• Covariate balance tables: compare pre-treatment characteristics of the treated unit and its synthetic counterpart
• RMSPE ratio: the ratio of post-to-pre-treatment RMSPE quantifies the magnitude of divergence after intervention

A quasi-experimental technique that estimates a treatment effect by comparing the average change over time in an outcome variable for a treatment group versus a control group. Unlike the Synthetic Control Method, DiD relies on a parallel trends assumption and typically uses a single unweighted control unit or aggregate. It is foundational for establishing causality in panel data where randomization is infeasible.

A statistical matching technique that estimates a treatment effect by accounting for the covariates that predict receiving the treatment. PSM reduces the dimensionality of matching to a single scalar score. While Synthetic Control constructs a counterfactual for a single treated unit over time, PSM is typically used in cross-sectional settings to pair treated units with similar control units based on observable characteristics.

An estimation method used to infer causal relationships from observational data by introducing an external instrument that affects the treatment but has no direct effect on the outcome. IV addresses endogeneity through a two-stage least squares process. This contrasts with the Synthetic Control Method, which handles unobserved confounders by constructing a data-driven counterfactual rather than relying on exclusion restrictions.

The cognitive and statistical process of imagining alternative scenarios and outcomes that would have occurred had specific prior actions or conditions been different. The Synthetic Control Method formalizes this by creating a weighted combination of untreated units to represent the missing counterfactual. Key components include:

• Factual World: The observed outcome with treatment
• Counterfactual World: The estimated outcome without treatment
• Causal Effect: The divergence between the two paths

The mean difference in outcomes between units assigned to a treatment and units assigned to a control, measuring the average causal impact across the entire population. The Synthetic Control Method focuses instead on the Average Treatment Effect on the Treated (ATT) for a single or small number of treated units, making it ideal for case studies where the intervention is unique, such as a policy change in one state or country.

An extraneous variable that influences both the dependent variable and independent variable, creating a spurious association that distorts the true causal effect. The Synthetic Control Method mitigates confounding by matching on pre-treatment outcomes and covariates, ensuring the synthetic unit replicates the trajectory of the treated unit before the intervention. This blocks backdoor paths without requiring explicit measurement of all confounders.