Synthetic control methods estimate a causal effect for a single treated unit by constructing a weighted combination of untreated donor units that approximates the treated unit’s pre-treatment trajectory. The treatment effect is the gap between the observed outcome and the synthetic counterfactual after treatment.

When to Use Synthetic Control vs. Other Methods

Feature	Cross-Sectional Event Study	Panel DiD	Synthetic Control
Number of treated units	Multiple firms	Multiple units	One unit
Control group	Market index	Untreated units	Weighted donors
Best for	Market reactions	Policy effects (many units)	Single policy case studies
Examples	M&A, earnings	Staggered regulations	Country-level reforms, single-state laws

When to use. Synthetic control is ideal when you have a single treated unit (one country, one state, one firm) and a pool of similar untreated units. Classic applications: the economic cost of conflict (Abadie & Gardeazabal 2003), California’s Proposition 99 (Abadie et al. 2010).

Method

Step 1: Construct the Synthetic Control

Find donor weights \(w_1, \ldots, w_J\) that minimize the pre-treatment distance between the treated unit and the weighted donors:

\[ \min_{w} \sum_{t=1}^{T_0} \left( Y_{1t} - \sum_{j=2}^{J+1} w_j Y_{jt} \right)^2 \quad \text{s.t.} \quad w_j \geq 0, \; \sum_{j} w_j = 1 \]

where \(Y_{1t}\) is the treated unit’s outcome, \(Y_{jt}\) are donor outcomes, and \(T_0\) is the last pre-treatment period.

Step 2: Estimate the Treatment Effect

The treatment effect at time \(t > T_0\) is the gap:

\[ \hat{\tau}_t = Y_{1t} - \sum_{j=2}^{J+1} \hat{w}_j Y_{jt} \]

Step 3: Inference via Placebo Tests

Placebo tests apply the synthetic control method to each donor unit in turn, pretending it was treated. The treated unit’s gap is compared to the distribution of placebo gaps.

\[ p\text{-value} = \frac{\text{rank of treated RMSPE among all RMSPEs}}{J + 1} \]

where RMSPE is the root mean squared prediction error in the post-treatment period.

Running in R

# Create synthetic control task
sc_task <- SyntheticControlTask$new(
  treated_data = treated,
  donor_data = donors,
  treatment_time = treatment_time
)

# Estimate synthetic control
result <- estimate_synthetic_control(sc_task)

# Donor weights
result$weights

NULL

# Trajectory plot: treated vs. synthetic
plot_synthetic_control(result, type = "trajectory")

# Gap plot: treatment effect over time
plot_synthetic_control(result, type = "gap")

Placebo Tests

# Run placebo tests (each donor as pseudo-treated)
placebo <- sc_placebo_test(sc_task)
plot_synthetic_control(placebo, type = "placebo")

A treated unit whose post-treatment gap is large relative to the placebo gaps provides evidence of a real treatment effect.

Diagnostic Checks

Check	What to Verify	If Failed
Pre-treatment fit	Synthetic control closely tracks treated unit before treatment	Add covariates or remove poor donors
Donor weights	Weights are not concentrated on a single unit	Expand donor pool
RMSPE ratio	Treated RMSPE >> placebo RMSPEs	Effect may not be significant
Leave-one-out	Results stable when removing individual donors	Check for outlier influence

Literature

Abadie, A. & Gardeazabal, J. (2003). The economic costs of conflict: A case study of the Basque Country. American Economic Review, 93(1), 113–132.
Abadie, A., Diamond, A. & Hainmueller, J. (2010). Synthetic control methods for comparative case studies. Journal of the American Statistical Association, 105(490), 493–505.
Abadie, A. (2021). Using synthetic controls: Feasibility, data requirements, and methodological aspects. Journal of Economic Literature, 59(2), 391–425.

Next Steps

Panel Event Studies — DiD estimators for multiple treated units
Expected Return Models — cross-sectional event study models
Inference & Robustness — wild bootstrap and multiple testing corrections