A Worked Synthetic Control Analysis for One Treated Region

From Donor Pool to Counterfactual Prediction

Synthetic control constructs a comparison for one treated region as a weighted average of untreated regions. The calculation is concrete, but its credibility depends on several linked choices: which regions can serve as donors, which pre-intervention features to match, how the weights are estimated, and whether the weighted donor outcomes provide a plausible prediction after treatment begins.

This tutorial works through those choices using hypothetical regional data. The aim is to calculate a synthetic control and its post-intervention predictions, not to treat a good numerical match as automatic proof of a causal effect. The identifying assumptions and donor-pool logic remain essential.

The donor-list index \(i\) is not a region label. Keeping that distinction explicit avoids confusion: the \(i\)th matrix column, weight \(w_i\), and donor outcome \(Y_{d_i,t}\) all refer to the same region, \(d_i\). The region label can be a name or a number; it need not equal \(i\).

Assumptions That Give the Calculation a Causal Role

Before calculating weights, specify the setting in which the synthetic control is intended to represent the treated region’s untreated outcome. Let \(T_0\) be the last pre-intervention period and \(T_0+1\) the first period with treatment. Write \(Y_{r,t}(0)\) for region \(r\)’s outcome at time \(t\) without the intervention, and \(Y_{r,t}(1)\) for its outcome under the intervention. The treated region’s post-intervention \(Y_{1,t}(0)\) is unobserved.

A useful synthetic control analysis requires, at minimum, that the intervention is not received by donor regions and does not affect their outcomes through spillovers; the treated region is not affected before the stated intervention date; and the outcome and predictor definitions are comparable across regions and time. It also requires that the weighted donors provide a credible approximation to the treated region’s untreated post-intervention path. Close pre-intervention predictor and outcome fit supports that approximation but cannot establish it by itself: a new, treated-region-specific shock after \(T_0\), for example, could break the prediction.

The convex-weight restriction means the synthetic control is a weighted average of the selected donors, rather than an extrapolation outside their range. If the treated region is far outside the donors’ pre-intervention predictor combinations, good fit may be impossible. Donor selection and predictor construction therefore come before interpreting any post-intervention calculation.

Worked Example: Screening the Donor Pool

Constructing Comparable Predictors

Predictors should be chosen using pre-intervention information and defined in the same way for the treated region and every donor. They may include pre-intervention outcomes, covariates, or summaries of outcome history. Averages over a specified window can reduce noise, while selected pre-intervention annual outcomes can preserve features of the trajectory. The window and definitions should be set so that they do not inadvertently include post-treatment information.

Predictors may be measured in different units. Since scale affects a weighted distance, the scaling rule and predictor-importance matrix should be stated rather than left implicit. In the example, each predictor is divided by a fixed, prespecified scale: ten outcome-index points for the pre-period mean, two percentage points for unemployment, and ten percentage points for manufacturing share. These are unit conversions, not standardizations estimated from the observed sample.

Estimating the Weights

The standard synthetic control weight problem chooses a convex combination of donors to match the treated region’s predictors. For a specified positive semidefinite matrix \(V\), the optimization is

The matrix \(V\) determines the relative importance of predictor mismatches. It can, for example, place greater emphasis on selected predictors or account for their scales. Here the predictors have already been rescaled by the stated fixed divisors, so use \(V=I\), the identity matrix. This gives equal weight to squared mismatches on the rescaled predictor scales. In an applied analysis, the predictor scales and importance choices should be justified and should not be selected simply to manufacture a preferred post-intervention result.

Predicting the Treated Region’s Untreated Outcomes

Once weights are fixed using pre-intervention information, the synthetic control’s outcome in period \(t\) is the same weighted average of the donor outcomes in that period. With the donor-list notation used throughout, the prediction for the treated region’s outcome without the intervention is

This is a counterfactual prediction, not an observed value. Its causal interpretation relies on the assumptions described earlier: in particular, that the selected weighted donors continue to approximate the treated region’s untreated outcome after the intervention begins.

What the Worked Calculation Does—and Does Not—Establish

The example has exact matching on three constructed predictors, and the estimated weights are transparent. Those facts are useful, but they do not establish that the post-intervention predictions are correct. The result depends on the substantive validity of the donors, the timing of treatment, the absence of donor contamination and spillovers, the pre-period information selected, and the plausibility that the weighted donors track A’s untreated path after 2020.

A further limitation is that a synthetic control is constrained to the convex hull of the selected donors. If no nonnegative weights summing to one can reproduce the treated predictors, the optimization returns the closest feasible match rather than an exact one. In that case the residual mismatch should be reported and considered when judging the prediction. A numerical optimizer’s output is not itself evidence that the design is credible.