Keeping spillover-affected countries in synthetic controls

When the best comparison is also affected

Synthetic control methods estimate an intervention’s effect by constructing a weighted combination of comparison units that resembles the treated unit before the intervention. Afterwards, the gap between the observed outcome and this synthetic counterpart estimates what the intervention changed.

The difficulty is that comparison units may also experience the intervention’s consequences. Keeping them can contaminate the estimate; excluding them can undermine the historical match. In their preprint, The inclusive synthetic control method, Roberta Di Stefano and Giovanni Mellace introduce a correction designed to address this trade-off.

Their motivating example is German reunification. Austria receives the largest weight in a conventional synthetic West Germany, but reunification could also have affected Austria. Demand and investment redirected towards East Germany are a possible channel discussed in the paper, not a mechanism established by the analysis.

Correcting linked estimates together

The inclusive synthetic control method, or iSCM, exploits weights estimated entirely from pre-intervention information. Researchers construct a synthetic control for the main treated unit and for every other unit designated as potentially affected. Those units can enter one another’s comparison pools.

Each resulting outcome gap combines the effect on the unit being studied with weighted effects on its affected donors, plus an approximation error. Because the donor weights are known, these relationships form a system of linear equations. Solving it jointly produces corrected estimates of the direct and spillover effects.

The distinction between contamination and approximation is essential. When the system is invertible and the assumed exposure classification is correct, the correction removes treatment-effect contamination exactly. It does not eliminate errors in the underlying synthetic controls. Those errors remain and can be amplified by the correction. The paper’s consistency result additionally requires approximation errors to vanish and the system to remain well conditioned as the pre-intervention period grows.

The effects themselves need not be uniform across units or time. However, researchers must specify which units could be affected, assume the remaining controls are unaffected, and assume no intervention effects occur before the intervention.

Why not simply exclude affected donors?

The authors show that the corrected counterfactual can ultimately be expressed using only unaffected controls’ outcomes. Its weights nevertheless incorporate information from the unrestricted pre-intervention fits.

That does not generally make it equivalent to excluding affected donors at the outset. Changing the donor pool changes the optimization problem. In the original method, researchers optimize predictor balance rather than directly minimizing the outcome’s historical prediction error, so removing donors can produce substantially different weights and fit.

There is no universal winner. When excluding potentially affected donors preserves a comparable fit, the authors favour that restricted approach. When affected units are important for fit, iSCM may help—but strong, mutually reinforcing weights between affected units can make the correction unstable.

A simplified version builds synthetic controls for the other affected units using unaffected donors only. This avoids the inversion-related amplification, but may sacrifice fit. The authors recommend reporting all four specifications: unrestricted, restricted, full inclusive and simplified inclusive.

Revisiting German reunification

The application uses data for West Germany, Austria and 15 other OECD countries over 1960–2003, with the post-intervention period beginning in 1990. The outcome is real GDP per capita at purchasing power parity, measured in 2002 US dollars.

Austria receives a weight of 0.42 in synthetic West Germany. Excluding it raises West Germany’s pre-intervention root mean squared prediction error from 119.07 to 260.74. A held-out pre-intervention validation exercise also favours retaining Austria. Diagnostics for Austria support retaining West Germany in its donor pool, and the estimated cross-weights imply a modest amplification factor of about 1.16.

The full iSCM estimate averages $1,677 less GDP per capita per year over 1990–2003, relative to the estimated no-reunification counterfactual. The unrestricted estimate is $1,570 less, while excluding Austria produces $2,387 less. These are estimated annual differences in GDP-per-capita levels, not annual growth-rate reductions. The restricted estimate is about 42% larger in magnitude than the full inclusive estimate.

Uncertainty remains consequential

For Austria, iSCM estimates an average negative spillover of $254 per person per year. That estimate cannot be distinguished from placebo variation: Austria’s placebo p-value is 15/16.

West Germany’s statistic is the largest in the comparison distribution, yielding 1/16, or approximately 0.06—the smallest attainable value with 15 unaffected controls. The authors describe their placebo procedure as a diagnostic, not a test with formal size guarantees. Correcting outcomes creates dependence across placebo statistics, amplification can favour rejection, and data-driven specification selection is not accounted for.

The contribution is therefore a way to remove a specific source of contamination while making the remaining fitting errors and stability trade-offs explicit—not a guarantee that the corrected historical estimates recover the true effects.