Identification Assumptions for Synthetic Control

Why Identification Needs More Than a Good Match

A synthetic comparison estimates what would have happened to an intervened unit by combining outcomes from units that were not treated. The central causal question is not whether this weighted combination resembles the treated unit before intervention. It is whether that same combination represents the treated unit’s untreated potential outcome after intervention. A close pre-intervention fit is useful evidence, but it does not alone establish that the relationship continues.

Let unit 1 be the treated unit and units $j=2,\ldots,J+1$ be donor units. Let $T_0$ be the last pre-intervention period. Write $Y_{it}(0)$ for unit $i$’s outcome at time $t$ under no intervention, and $Y_{it}(1)$ for its outcome under the intervention. Assume unit 1 receives the intervention after $T_0$, while the donors are not directly assigned to it. The effect on unit 1 at a post-intervention time $t$ is $\tau_{1t}=Y_{1t}(1)-Y_{1t}(0)$. An average effect over a set $\mathcal{T}_{post}$ of post-intervention periods is the average of these period-specific effects.

Definition: Given weights $w_j$, the observed weighted donor outcome at time $t$ is $\sum_{j=2}^{J+1}w_jY_{jt}$. The synthetic untreated outcome is $\sum_{j=2}^{J+1}w_jY_{jt}(0)$. These are equal exactly when $\sum_{j=2}^{J+1}w_j\bigl[Y_{jt}-Y_{jt}(0)\bigr]=0$; no spillovers for positively weighted donors, which makes each such donor’s observed outcome equal its outcome under no intervention, is a sufficient condition. A synthetic estimate of unit 1’s treatment effect is the treated unit’s observed post-intervention outcome minus the observed weighted donor outcome, subject to the identification assumptions developed below.

A common synthetic-control specification uses nonnegative weights that sum to one: $w_j\geq 0$ and $\sum_j w_j=1$. This restriction makes the synthetic unit a weighted average rather than an extrapolation beyond the donor outcomes. The identification argument below applies to fixed weights; the weights are chosen using pre-intervention information and then held fixed when evaluating post-intervention outcomes.

The Conditions for Identification

Three requirements do distinct jobs. No spillovers allow observed donor outcomes to stand in for donors’ untreated outcomes. A stable untreated-outcome relationship makes the weighted donors represent unit 1 after intervention. Adequate pre-intervention fit provides evidence that this relationship describes the pre-intervention data. Donor support concerns whether a suitable weighted representation is available in the first place.

No spillovers: For every positively weighted donor and relevant post-intervention time, the intervention on unit 1 does not change that donor’s outcome. Thus $Y_{jt}=Y_{jt}(0)$ for those donors and times. This condition rules out effects transmitted through trade, migration, shared services, policy imitation, or other channels. It is not enough that donors were not formally assigned the intervention.

For the pre-intervention period, also require no anticipation: before the intervention begins, the treated unit’s observed outcome equals its untreated potential outcome. This lets the observed pre-intervention record inform the relationship among untreated outcomes. If units respond in advance to a known intervention, pre-intervention outcomes may already reflect treatment-related changes.

Stable untreated-outcome relationship: For the chosen weights, the treated unit’s untreated outcome is represented by the same weighted combination of donor untreated outcomes in the post-intervention periods: $$ Y_{1t}(0)=\sum_{j=2}^{J+1}w_jY_{jt}(0), \qquad t\in\mathcal{T}_{post}. $$ This is the central counterfactual restriction. It need not hold by definition merely because the weights fit pre-intervention outcomes well; it must be justified as a relationship that remains valid after intervention.

In practice, exact equality is an idealization. Define the untreated-outcome discrepancy, or prediction error, as $r_t=Y_{1t}(0)-\sum_j w_jY_{jt}(0)$. If $r_t$ is not zero, the synthetic estimate does not identify the period-specific effect exactly. The discrepancy may be small enough for an approximation to be useful, but it is then a source of bias rather than a random sampling error that the comparison automatically removes.

A transparent sufficient model illustrates what stability entails. Suppose untreated outcomes can be written as $Y_{it}(0)=\delta_t+\mu_i'f_t+e_{it}$, where $\delta_t$ is a time-specific component shared by units, $f_t$ is a vector of time-varying factors, $\mu_i$ is unit $i$’s factor loading, and $e_{it}$ is a remaining unit-time component. If the weights sum to one and match the treated unit’s loadings, $\mu_1=\sum_j w_j\mu_j$, then the common component and factor component match. Exact representation additionally requires $e_{1t}=\sum_jw_je_{jt}$ in the post period. Thus matching the stable components alone is not a guarantee if unrepresented shocks remain. Adequate pre-fit can support the plausibility of such matching, but cannot prove that latent loadings or residual relationships remain aligned after intervention.

Donor support: The treated unit must be representable by the donors with the permitted weights on the characteristics or stable outcome components relevant to untreated outcomes. With nonnegative weights summing to one, exact matching requires the treated unit’s relevant characteristics to lie in the donors’ convex hull. If it lies outside that hull, no convex weighted donor combination can match it exactly; the analyst must accept mismatch or use extrapolative weights, which requires stronger justification.

Deriving the Causal Comparison

Let the observed outcome of unit 1 in a post-intervention period be $Y_{1t}(1)$, and suppose the treated unit is observed under the intervention. The observed comparison is $Y_{1t}(1)-\sum_j w_jY_{jt}$. Under no spillovers, the donor observations equal their untreated potential outcomes. Subtracting and adding the treated unit’s untreated outcome gives the exact decomposition below.

\begin{aligned} Y_{1t}(1)-\sum_j w_jY_{jt} &=Y_{1t}(1)-\sum_j w_jY_{jt}(0)\\ &=\bigl[Y_{1t}(1)-Y_{1t}(0)\bigr] +\bigl[Y_{1t}(0)-\sum_j w_jY_{jt}(0)\bigr]\\ &=\tau_{1t}+r_t. \end{aligned}

This derivation shows exactly what the assumptions buy. No spillovers justify replacing observed donor outcomes by untreated donor outcomes. Stable representation sets $r_t=0$, so the observed gap equals the causal effect $\tau_{1t}$. If the relationship is only approximate, the gap equals the effect plus the untreated prediction error. For an average over post-intervention periods, the same decomposition holds after averaging: the average observed gap equals the average effect plus the average prediction error.

Adequate pre-intervention fit concerns observed data from before treatment. Under no anticipation, those observations are untreated outcomes, so close fit indicates that $r_t$ was small in the pre-period. It does not force $r_t=0$ after $T_0$. A new shock affecting the treated unit differently, a change in the factor relationship, or an intervention spillover to donors can break the required post-period comparison even when the pre-period fit is excellent.

Worked Examples

Worked Example: An Exact Counterfactual Match

Suppose a hypothetical region is treated after period 2. Two untreated donor regions receive weights $w_2=0.6$ and $w_3=0.4$. In period 2, their outcomes are 40 and 65, while the treated region’s outcome is 50. In post-period 3, the observed donor outcomes are 44 and 70, and the treated region’s observed outcome under intervention is 61. Assume no anticipation, no spillovers, and that the weighted untreated relationship remains exact in period 3.

The weights are nonnegative and sum to one: $0.6+0.4=1$. The pre-period weighted outcome is $0.6(40)+0.4(65)=24+26=50$, exactly matching the treated region’s pre-intervention outcome. Under no spillovers, the post-period donor outcomes are their untreated outcomes, so the synthetic untreated outcome is $0.6(44)+0.4(70)=26.4+28=54.4$.

By the stable-relationship assumption, $Y_{1,3}(0)=54.4$. The estimated effect is therefore $61-54.4=6.6$ outcome units. Here the exact pre-fit is consistent with the counterfactual representation, but the post-period conclusion also relies on the stated stability and no-spillover assumptions; it does not follow from the pre-fit alone.

Worked Example: A Small Pre-Fit Error Does Not Guarantee Stability

In a second hypothetical comparison, the weighted donors’ outcome is 99.5 in the last pre-intervention period, while the treated unit’s outcome is 99. The pre-period discrepancy is $r_{pre}=99-99.5=-0.5$ units, a close fit. After intervention, suppose the observed weighted donor outcome is 120 and the treated unit’s observed outcome is 130. Assume no spillovers, but suppose a treated-unit-specific untreated shock would have raised its untreated outcome to 124.

The post-period discrepancy is $r_{post}=124-120=4$ units. The actual treatment effect in this hypothetical example is $130-124=6$ units. But the observed treated-minus-synthetic gap is $130-120=10$ units. The decomposition verifies the difference: $10=6+4$. Thus the estimate overstates the effect by 4 units, despite the pre-intervention discrepancy being only 0.5 units in magnitude. The example isolates why pre-fit is evidence about earlier periods, not a proof of post-period stability.

Worked Example: Spillovers Contaminate the Observed Donor Combination

Suppose a hypothetical intervention would leave two donors’ untreated post-period outcomes at 30 and 50. Their weights are $0.5$ each, so their synthetic untreated outcome is $0.5(30)+0.5(50)=40$. The treated unit’s untreated outcome is also 40, and its observed outcome under intervention is 48. The true effect is $48-40=8$ units.

Now suppose the intervention spills over to the donors, raising their observed outcomes by 4 and 2 units. The observed donor outcomes are then 34 and 52, and the observed weighted donor outcome is $0.5(34)+0.5(52)=17+26=43$. The observed comparison is $48-43=5$, which is 3 units below the true effect of 8. The donor outcomes are not the donors’ untreated outcomes, so the no-spillover step in the identification argument fails. A weighted combination of observed donors cannot be called a synthetic untreated outcome in this situation without a separate adjustment or argument.

Worked Example: Checking Convex-Hull Support

Consider one relevant scalar characteristic that determines untreated outcomes through a stable relationship. The treated unit has characteristic value 2.5, and two donors have values 1 and 4. With equal weights, the synthetic characteristic is $0.5(1)+0.5(4)=2.5$, so the treated value is inside the donors’ convex hull and is exactly matched.

If instead the donor values are 1 and 2, every nonnegative weighted average summing to one lies between 1 and 2. In particular, $w(1)+(1-w)(2)=2-w\leq 2$ for $0\leq w\leq1$, so it cannot equal 2.5. A linear combination can reach 2.5 only by allowing a negative weight: solving $w(1)+(1-w)(2)=2.5$ gives $2-w=2.5$, hence $w=-0.5$ and the other weight $1.5$. That is extrapolation, not a convex combination. It may be possible under a correctly specified linear relationship, but the data do not provide the same support-based reassurance as an in-hull match.

Common Mistakes and Practical Meaning

Calling the observed donor sum the untreated counterfactual without qualification. Observed donors can be affected by spillovers. Distinguish $\sum_jw_jY_{jt}$ from $\sum_jw_jY_{jt}(0)$, and equate them only under no spillovers for the weighted donors.
Treating a good pre-fit as proof of identification. A good pre-fit shows similarity in the periods observed before intervention. Identification also needs the untreated relationship to remain stable in the post-period.
Ignoring prediction error in the causal gap. The observed comparison equals $\tau_{1t}+r_t$, not automatically $\tau_{1t}$. If the synthetic untreated outcome misses the treated unit’s counterfactual by 4 units, the effect comparison carries that 4-unit discrepancy.
Assuming every treated unit is supported by the donors. A convex weighted average cannot match characteristics outside the donors’ convex hull. A close outcome fit on a short pre-period does not erase that limitation.

These assumptions matter because synthetic control is a counterfactual design, not merely a method for drawing a weighted trend. No spillovers protect the donor outcomes from treatment contamination; stable relationships connect those donors to the treated unit’s missing outcomes; and donor support determines whether the proposed weighted representation is credible. Pre-intervention fit is a diagnostic for the historical part of this argument, while the post-intervention counterfactual ultimately depends on assumptions that cannot be directly observed.

Check Your Understanding

Answer each question by stating the relevant assumption or calculation.

Why is the observed weighted donor outcome not automatically the synthetic untreated outcome after intervention?
Write the decomposition of the observed treated-minus-donor gap into a treatment effect and an untreated-outcome discrepancy.
A synthetic comparison fits the treated unit almost exactly before intervention. What additional condition is needed for the same weights to identify the post-intervention untreated outcome?
With nonnegative weights summing to one, what does it mean for the treated unit to lie outside the donors’ convex hull?
If the true post-period effect is 7 units and the untreated prediction error is $-2$ units, what is the observed treated-minus-synthetic gap? Explain the sign of the prediction error’s contribution.