Synthetic Control Weights and the Counterfactual

From Untreated Units to a Counterfactual

When one region, firm, or other aggregate unit receives an intervention, its post-intervention outcome cannot be compared directly with its own outcome in the same period absent treatment: that counterfactual is unobserved. Synthetic control constructs a comparison unit from untreated units, called donors. The weights are chosen so that the weighted donors resemble the treated unit on selected pre-intervention predictors, often including its pre-intervention outcome path.

The central idea is simple, but its interpretation requires care. A weighted donor combination is a model-based estimate of what the treated unit’s outcome would have been without intervention. Matching predictors is a way to construct that estimate; matching alone does not prove that it is the correct counterfactual.

Under no anticipation, the treated unit's observed outcome before intervention is \(Y_{1t}(0)\), and its observed outcome after intervention is \(Y_{1t}(1)\). If the weighted donor outcome represents \(Y_{1t}(0)\) after intervention, the gap estimates the treated unit's effect in that period. This is an effect for the treated unit, not automatically an average effect for a wider population.

Choosing Weights by Predictor Matching

Collect \(K\) predictors for the treated unit in a vector \(X_1\), and arrange the corresponding donor predictors as columns of a \(K \times J\) matrix \(X_0\). Predictors may include pre-intervention outcome summaries and other pre-intervention characteristics. A vector \(W=(w_2,\ldots,w_{J+1})'\) specifies the donor weights. Given a fixed positive semidefinite weighting matrix \(V\), one standard synthetic control choice minimizes the weighted discrepancy between the treated predictors and the weighted donor predictors:

The matrix \(V\) determines the relative importance of predictor discrepancies in this objective. The weight restrictions make the synthetic control a convex combination: it cannot assign negative shares or have a total weight other than one. Consequently, its predictor values lie within the convex hull of donor predictor vectors. The restriction can make the comparison easier to interpret, but it also means that an exact match may be impossible when the treated unit lies outside that hull.

Once the weights are chosen, apply the same weights to donor outcomes in every period. The post-intervention estimate of the treated unit’s untreated outcome and the resulting gap are:

If the treated unit is exposed to treatment in period \(t\), then \(Y_{1t}=Y_{1t}(1)\), so the gap estimates \(Y_{1t}(1)-Y_{1t}(0)\) when the synthetic outcome is a valid estimate of \(Y_{1t}(0)\). A positive gap means the observed treated-unit outcome exceeds its estimated untreated outcome; a negative gap means it falls below. The units are those of the outcome.

Why the Gap Has a Counterfactual Interpretation

The arithmetic behind the estimate is immediate; the causal claim is not. The gap can be interpreted as an effect only if the weighted donor outcomes provide a credible estimate of the treated unit’s untreated post-intervention outcomes. The design therefore depends on the intervention and comparison being well defined, on donors remaining untreated and unaffected by the intervention, and on pre-intervention information being sufficient to construct a useful comparison.

A common justification is that the treated unit and its synthetic counterpart share the relevant untreated outcome-generating features, so that matching their pre-intervention predictors also produces a credible post-intervention counterfactual. This is a substantive assumption, not a consequence of the minimization problem. In particular, a low predictor discrepancy cannot rule out a post-intervention shock unique to the treated unit. Likewise, an intervention that affects donors through spillovers undermines their role as untreated comparisons.

Worked Examples

A Short Derivation of the Gap Estimator

For a post-intervention period \(t\), define the treated unit’s effect as \(\tau_{1t}=Y_{1t}(1)-Y_{1t}(0)\). The observed treated outcome is \(Y_{1t}=Y_{1t}(1)\). If the synthetic outcome estimates the missing untreated outcome, then replacing \(Y_{1t}(0)\) by \(\widehat Y_{1t}(0)\) yields the estimator:

This derivation separates two parts of the analysis. The observed treated outcome is data; the weighted donor outcome is the estimated counterfactual. The difference is an effect estimate only insofar as that estimated counterfactual is credible. The optimization supplies the weights, not a proof of the required counterfactual relationship.

Common Mistakes and Why Synthetic Control Matters

• Calling the gap causal by definition. A post-intervention difference is an estimate of an effect only under assumptions that make the synthetic outcome a credible untreated counterfactual.
• Using weights that do not satisfy the constraints. Negative weights or weights that do not sum to one are not the convex synthetic control defined here. They can create extrapolations outside the donors’ predictor range.
• Assuming pre-intervention fit guarantees post-intervention fit. Matching observed predictors is useful evidence about comparability, but does not exclude a treated-unit-specific shock after intervention.
• Interpreting the gap as a population average. The basic construction estimates effects for the treated unit over time. Generalizing beyond that unit requires additional justification.
• Ignoring interference or donor treatment. If donors are affected by the intervention, their outcomes may no longer represent untreated outcomes.

Synthetic control is useful when a single treated aggregate unit has no obvious single comparison unit, but a set of untreated units can be combined to resemble it before intervention. Its transparency comes from seeing which donors contribute and how their outcomes are combined. Its credibility depends on both the quality of the pre-intervention match and the assumptions connecting that match to the unobserved post-intervention counterfactual.