1 Introduction
What would California's per-capita cigarette consumption have been if Proposition 99- the 1988 tobacco tax and control programme had never passed? This is a counterfactual question: we observe one path (Proposition 99 was enacted) but not the other (what would have happened without it).
The standard difference-in-differences answer is to compare California to some control state. But which state? No single state looks like California on the relevant pre-treatment predictors income, smoking prevalence, population, retail prices. Averaging over all other states dilutes the comparison.
Abadie et al. [2010] introduced the synthetic control method to address exactly this problem. Instead of choosing a single control unit or a simple average, the method constructs a weighted combination of control units a "synthetic" California that closely resembles the treated unit in pre-treatment periods. The counterfactual is then what the synthetic unit does after treatment.
The method has become one of the most influential tools in comparative case study analysis. This article explains the mechanics, assumptions, inference procedure, and key applications.
2 The Setup
Consider $J+1$ units indexed $j=0,1,...,J$ Unit $j=0$ is the treated unit; units $j=1,...,J$ form the donor pool of potential controls. There are T periods, and treatment begins at period $T_{0}+1$, so periods $1,...,T_{0}$ are pre-treatment.
Let $Y_{jt}^{N}$ denote the potential outcome for unit j at time t in the absence of treatment, and $Y_{jt}^{I}$ the potential outcome under treatment. We observe:
$Y_{jt}=Y_{jt}^{N}+\alpha_{jt}D_{jt}$
where $D_{jt}=1$ if unit j is treated at time t, and $\alpha_{jt}$ is the unit-period treatment effect. The estimand is $\alpha_{0t}=Y_{0t}^{I}-Y_{0t}^{N}$ for $t>T_{0}$. We observe $Y_{0t}^{I}$ (the treated unit's outcome post-treatment). The challenge is to estimate $Y_{0t}^{N}$ the counterfactual.
3 Constructing the Synthetic Control
The synthetic control is a vector of weights $w=(w_{1},...,w_{J})$ with $w_{j}\ge0$ and $\sum_{j}w_{j}=1$ chosen so that the weighted combination of donor units matches the treated unit's pre-treatment characteristics.
Define the pre-treatment predictor vector for unit jas $X_{j}=(X_{1j},...,X_{Kj})^{\prime}$, which includes pre-treatment outcome values and other covariates. The optimal weights solve:
$min_{w}(X_{0}-X_{W})^{\prime}V(X_{0}-X_{W})$ s.t. $w_{j}\ge0$, $\sum_{j}w_{j}=1$
(1)
where $X_{W}=\sum_{j}w_{j}X_{j}$ is the synthetic control's predictor vector and V is a positive semi-definite matrix that weights the predictors by their importance.
The choice of V matters: Abadie et al. [2010] recommend choosing V to minimise the pre-treatment mean squared prediction error (MSPE) of the synthetic control effectively letting the data determine which predictors are most relevant for constructing the counterfactual.
Once the optimal $w^{*}$ is obtained, the estimated treatment effect in post-treatment period t is:
$\hat{\alpha}_{0t}=Y_{0t}-\sum_{j=1}^{J}w_{j}^{*}Y_{jt}$
(2)
4 The Key Identifying Assumption
The synthetic control estimator is consistent for $\alpha_{0t}$ under the assumption that there exists a set of weights $w^{*}$ such that the synthetic control exactly replicates the treated unit's pre-treatment outcome path and covariates. Formally, Abadie et al. [2010] assume a factor model for potential outcomes:
$Y_{jt}^{N}=\delta_{t}+\theta_{t}^{\prime}Z_{j}+\lambda_{t}^{\prime}\mu_{j}+\epsilon_{jt}$
(3)
where $\delta_{t}$ is a common time trend, $Z_{j}$ are observed covariates, $\lambda_{t}$ are unobserved common factors, $\mu_{j}$ are unit-specific factor loadings, and $\epsilon_{jt}$ is idiosyncratic noise.
Under this model, if the synthetic control matches the treated unit on pre-treatment outcomes and covariates, then the weighted combination of donor units also matches the unobserved factors $\lambda_{t}^{\prime}\mu_{j}$ even in post-treatment periods. This is the key insight: good pre-treatment fit implies a good counterfactual.
The assumption is more transparent than parallel trends in standard DiD: the pre-treatment fit is observable and can be directly assessed by the researcher.
5 Inference: Permutation-Based Placebo Tests
Standard asymptotic inference is not available for synthetic control because the number of treated units is typically one (or very small). Abadie et al. [2010] develop a permutation-based inference procedure. The procedure is:
- Apply the synthetic control method to each donor unit $j\in\{1,...,J\}$ using the remaining units as the donor pool. This generates J "placebo" treatment effects $\hat{\alpha}_{jt}$.
- The p-value for the null of no treatment effect is the proportion of placebo units with a post-treatment MSPE ratio $(post/pre)$ as large as the treated unit's ratio.
Placebo units with poor pre-treatment fit should be discarded before computing p-values, since they provide little information about the null distribution. The inference is valid in the sense of randomisation inference under the sharp null.
6 The California Tobacco Application
The canonical application estimates the effect of California's Proposition 99 on per-capita cigarette sales [Abadie et al., 2010]. Using data from 38 donor states from 1970 to 2000 (Proposition 99 passed in 1988), the synthetic California is a weighted combination of Colorado, Connecticut, Montana, Nevada, and Utah.
The synthetic control closely tracks California's pre-treatment cigarette consumption. After 1988, California's actual consumption falls substantially below the synthetic control, implying a treatment effect of approximately -25 packs per capita per year by 2000-a large and statistically significant effect confirmed by the placebo tests.
7 Extensions and Limitations
7.1 The Augmented Synthetic Control (ASCM)
A key limitation of the original synthetic control is that perfect pre-treatment balance is rarely achieved. When there is residual pre-treatment imbalance, the post-treatment comparison is biased. Ben-Michael et al. [2021] propose the Augmented Synthetic Control (ASCM), which adds a bias-correction term estimated via an outcome model. ASCM reduces to the original synthetic control when pre-treatment fit is perfect, but improves on it when balance is imperfect.
7.2 Synthetic DiD
Arkhangelsky et al. [2021] combine the synthetic control idea with difference-in-differences, constructing both unit weights (as in SC) and time weights (reweighting pre-treatment periods) to achieve doubly-robust estimation.
7.3 Limitations
The method has known weaknesses:
- Interpolation bias: The synthetic control is a convex combination of donor units. If the treated unit is an extrapolation (outside the convex hull of donor units), the weights will not replicate the treated unit well regardless of optimisation.
- Donor pool selection: The choice of which units to include in the donor pool is consequential and somewhat arbitrary. Including units that were themselves affected by the treatment ("spillover contamination") biases the counterfactual.
- Single treated unit: Power is limited when J is small, as the permutation distribution has few points.
8 When to Use Synthetic Control
The synthetic control method is most appropriate when:
- There is a single (or small number of) treated unit(s).
- The pre-treatment period is long enough to assess and validate the pre-treatment fit.
- A suitable donor pool of comparable, unaffected units exists.
- The treatment is a large, discrete policy change at a specific point in time.
It is not appropriate when the treated unit is a clear extrapolation from the donor pool, when the pre-treatment period is short, or when many units receive treatment simultaneously (DiD or staggered DiD methods are more appropriate then).
9 Conclusion
The synthetic control method is a powerful tool for comparative case study analysis. By constructing a data-driven counterfactual rather than relying on a single comparator or an unweighted average, it improves on traditional DiD for settings with one or few treated units. The transparency of pre-treatment fit makes the key assumption more verifiable than parallel trends. Combined with permutation-based inference and modern extensions like ASCM, the method has become essential in the applied econometrician's toolkit.
References
- 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.
- Abadie, A. and Gardeazabal, J. (2003). The economic costs of conflict: A case study of the Basque Country. American Economic Review, 93(1):113-132.
- Ben-Michael, E., Feller, A., and Rothstein, J. (2021). The augmented synthetic control method. Journal of the American Statistical Association, 116(536):1789-1803.
- 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.
- Abadie, A. (2021). Using synthetic controls: Feasibility, data requirements, and methodological aspects. Journal of Economic Literature, 59(2):391-425.