1 The Assumption Hiding in Plain Sight

The synthetic control method (SCM) of Abadie et al. [2010] is one of the great success stories of modern policy evaluation. Confronted with a single treated unit a state that raised its cigarette tax, a country that suffered a terrorist campaign SCM builds a "synthetic" counterfactual as a weighted average of untreated units chosen to match the treated unit's pre-treatment outcome path. Where the pre-treatment fit is good, the post-treatment gap between the treated unit and its synthetic twin is read as the treatment effect.

The method's credibility rests on an assumption that is rarely stated explicitly: that a perfect synthetic control exists. That is, there is a set of weights under which the synthetic unit's untreated potential outcomes equal the treated unit's, at least in expectation. Powell [2026] argues that this assumption is far stronger than it looks and frequently false. If outcomes contain transitory shocks with nonzero asymptotic variance (idiosyncratic noise that does not average away), no fixed weighting of control units can reproduce the treated unit's realised path, because the noise in the treated unit is its own. Worse, a perfect synthetic control may fail to exist even in expectation: if the treated unit's factor loadings lie outside the convex hull of the donors', the weighted average can never match it on average. When the perfect-control assumption fails, the standard estimator is biased and its permutation-based inference is miscalibrated.  

2 What Goes Wrong, Concretely

Adopt the now-standard factor (interactive fixed effects) model for untreated potential outcomes,

where λᵢ are unit-specific factor loadings, Fₜ are common time factors, and ϵᵢₜ is a transitory shock. SCM weights {wᵢ} seek to reproduce the treated unit (unit 1) so that ∑ᵢ≠₁ wᵢλᵢ = λ₁. Two problems arise:  

1. Transitory shocks do not cancel. Even if the loadings match perfectly, the synthetic control reproduces ∑ᵢ≠₁ wᵢϵᵢₜ, not ϵ₁ₜ. Matching the treated unit's realised pre-period path therefore forces the weights to chase noise, biasing the post-period counterfactual. Good pre-treatment fit can be a symptom of overfitting to ϵ₁ₜ, not a sign of validity.  

2. The convex hull may not contain the target. If λ₁ lies outside the convex hull of {λᵢ}ᵢ≠₁, the interpolation bias is irreducible: no nonnegative weights summing to one can match the treated unit even on average. Abadie et al. [2010] acknowledged this, but standard practice often ignores it.  

3 The Imperfect Synthetic Control Estimator

Powell [2026] reorganises the problem around three ideas.  

1. Synthesise every unit, not just the treated one. Rather than building a single synthetic control for the treated unit, the procedure constructs a synthetic control for each unit in the sample, treated and untreated alike. Each unit thus yields its own treated-minus-synthetic gap. For untreated units these gaps are pseudo-treatment effects whose true value is zero; for the treated unit the gap estimates the effect of interest. This turns the entire donor pool into a reference distribution.  
2. Moment conditions robust to transitory shocks. Instead of demanding that the synthetic control match the treated unit's realised outcomes, Powell [2026] derives moment conditions that hold in expectation given the presence of ϵᵢₜ. The weights are chosen to satisfy these moments rather than to minimise in-sample pre-treatment distance, so the estimator does not overfit transitory noise. Schematically, the target estimand the average treatment effect on the treated, is identified by a moment of the form  

where the weights wᵢ⁎ solve a companion set of moment restrictions that purge the contribution of the idiosyncratic shocks rather than fitting them.  

3. A weighting metric that discards bad matches. Because some units genuinely have no good synthetic control (their loadings sit outside the donor hull), Powell [2026] introduces a data-driven weighting metric that asymptotically excludes units for which no appropriate synthetic control can be formed. Units that cannot be matched are down-weighted to zero, so the final estimand is defined over the subpopulation for which valid counterfactuals exist an honest restriction rather than a silent extrapolation.  

Inference With Few Controls

A perennial headache for SCM is inference. With one treated unit and a handful of donors, asymptotics in the cross-section are unavailable, and Abadie et al. [2010] rely on permutation (placebo) tests whose validity itself requires the perfect-control assumption. The imperfect-control construction sidesteps this. Because the method produces a synthetic control- and hence a gap estimate for every unit, it generates multiple estimates of the treatment effect from a single dataset. The dispersion of these estimates supplies a basis for inference that remains valid even when the number of control units is small. In effect, the placebo distribution is promoted from an informal diagnostic to the formal engine of standard errors and confidence intervals, and it no longer presumes that a perfect match exists.  

5 Application: Wisconsin's Handgun Waiting Period

Powell [2026] illustrates the method by evaluating the 2015 repeal of Wisconsin's 48-hour waiting period for handgun purchases, asking whether removing the cooling-off delay affected suicide rates. The setting is a hard one for classical SCM: there is a single treated state, a modest pool of donor states, and outcome series laden with year-to-year noise exactly the conditions under which the perfect-control assumption is least tenable and few-donor inference is most fragile. The imperfect synthetic control estimator delivers an effect estimate together with calibrated uncertainty that does not lean on a perfect pre-treatment match, providing more credible inference about the policy's consequences for suicide than a naive synthetic control would.  

6 Where It Fits

Imperfect synthetic controls belong to a broader 2020s movement to put single-treated-unit inference on firmer statistical ground, alongside synthetic difference-in-differences [Arkhangelsky et al., 2021], matrix completion [Athey et al., 2021], and conformal inference for SCM [Chernozhukov et al., 2021]. What distinguishes Powell [2026] is its frank acknowledgement that the object SCM has always promised a perfect counterfactual twin generally does not exist, and its reconstruction of the estimator and its inference around that fact. For practitioners the practical advice is bracing: stop treating excellent pre-treatment fit as a badge of validity, check whether the treated unit even lies inside the donor hull, and prefer inference procedures that survive the realistic case in which the match is imperfect. A method built to fail gracefully when no twin exists is, paradoxically, more trustworthy than one that always finds a twin.  

References

1. Abadie, A., Diamond, A., and Hainmueller, J. (2010). Synthetic control methods for comparative case studies: estimating the effect of California's tobacco control program. Journal of the American Statistical Association, 105(490), 493-505.  
2. Arkhangelsky, D., Athey, S., Hirshberg, D. A., Imbens, G. W., and Wager, S. (2021). Synthetic difference-in-differences. American Economic Review, 111(12), 4088-4118.  
3. Athey, S., Bayati, M., Doudchenko, N., Imbens, G., and Khosravi, K. (2021). Matrix completion methods for causal panel data models. Journal of the American Statistical Association, 116(536), 1716-1730.
4. Chernozhukov, V., Wüthrich, K., and Zhu, Y. (2021). An exact and robust conformal inference method for counterfactual and synthetic controls. Journal of the American Statistical Association, 116(536), 1849-1864.  
5. Powell, D. (2026). Imperfect synthetic controls. Journal of Applied Econometrics, forthcoming.