1 Motivation: The Problem of a Single Treated Unit

Difference-in-differences works well when you have many treated and many control units. But what if the treatment applies to a single unit a state, a country, a city? That is the situation Abadie et al. [2010] faced when studying the effect of California's Proposition 99 (a 1988 tobacco tax and anti-smoking campaign) on cigarette sales. California was the only treated unit.

You could compare California to the rest of the United States. But the rest of the US is not a good counterfactual for California: it differs in size, demographics, existing smoking rates, and dozens of other characteristics. You could compare California to a single similar state say, Nevada or Arizona. But no single state perfectly mirrors California's pre-treatment trajectory.

The synthetic control method solves this by constructing a weighted combination of control units a "synthetic California" whose pre-treatment characteristics match California's as closely as possible. This synthetic unit then serves as the counterfactual.

2 The Core Idea

The fundamental intuition is straightforward: instead of choosing one comparison unit, choose a combination of many units whose weighted average closely tracks the treated unit before treatment. If the synthetic control tracks the treated unit well in the pre-treatment period, then under the assumption that what produced this match would have continued- it tells us what would have happened to the treated unit in the absence of treatment.

Formally, let Yⱼₜᴺ denote the outcome (e.g. cigarette sales per capita) for unit j at time t in the absence of treatment, and let Y₁ₜᴵ denote the outcome for the treated unit (California, j=1) under treatment. The causal effect of treatment at time t is:

The problem is that Y₁ₜᴺ is not observed after treatment. The synthetic control estimator replaces Y₁ₜᴺ with:

where j=2,...,J+1 are the control units (the "donor pool") and wⱼ ≥ 0, ∑ wⱼ = 1 are weights chosen to minimise pre-treatment discrepancies.

3 How the Weights Are Chosen

The weights w=(w₂, ..., wⱼ₊₁) are chosen to minimise the pre-treatment distance between California and the synthetic control:

Here X₁ₛ are pre-treatment predictor values for the treated unit (average cigarette sales in 1980-1988, income, beer consumption, percentage of young people, etc.) and vₛ are predictor weights that reflect the importance of each predictor. The predictor weights v are often chosen to minimise the pre-treatment fit of the outcome variable itself.

Example. For California's Proposition 99, Abadie et al. [2010] find that a synthetic California constructed as approximately 55% Colorado, 24% Utah, 13% Nevada, and 8% New Mexico closely tracks California's cigarette sales from 1970-1988. After 1988, real California sales fall substantially below synthetic California, suggesting a treatment effect of about 25 fewer cigarette packs per capita per year by the mid-1990s.

4 Checking the Method: Permutation Inference

How do we know whether the estimated effect is real or due to chance? Standard statistical tests based on t-statistics do not apply here, because we have only one treated unit. Abadie et al. [2010] propose permutation inference (also called placebo tests):

1. Apply the synthetic control method to each control state, pretending it was the treated unit.
2. Compute the post-treatment "effect" for each placebo unit.
3. The true treatment effect is significant if it is large relative to the distribution of placebo effects.

The ratio:

measures how large the post-treatment divergence is relative to the pre-treatment fit quality. California has the largest ratio among all donor states, placing it at the tail of the permutation distribution. This provides evidence that the effect is not a statistical artefact.

5 Key Assumptions

The synthetic control estimator is valid under two main conditions:

1. Good pre-treatment fit. If the synthetic control closely tracks the treated unit in the pre-treatment period, it is plausible that it would have continued to do so absent treatment. Poor pre-treatment fit is a warning sign that the synthetic control may not be a valid counterfactual.
2. No anticipation. Treatment effects should not begin before the formal start of the intervention. Abadie et al. [2010] check this by plotting the pre-treatment gap between California and its synthetic control if there is no pre-treatment divergence, anticipation effects are unlikely.

An additional practical condition is that the treated unit's pre-treatment characteristics must lie in the convex hull of the donor pool. If the treated unit is extreme on some characteristic it is much richer, or has much higher baseline outcomes, than all donors- the weighted average can never approximate it, and the synthetic control will extrapolate outside the observed data ("extrapolation bias").

6 Comparison to Difference-in-Differences

Neither method dominates the other. Synthetic control is ideal when there is a single treated unit with a long pre-treatment series. DiD is better suited to multiple treated units and shorter panels.‍


7 Common Mistakes

1. Ignoring pre-treatment fit quality. A synthetic control with poor pre-treatment fit is not a valid counterfactual. Always plot the pre-treatment gap and report the pre-treatment MSPE.
2. Choosing the donor pool after seeing the results. The donor pool must be chosen on substantive grounds before examining post-treatment outcomes. Cherry-picking donors to produce a desired result is a form of specification searching.
3. Using too few pre-treatment periods. With only one or two pre-treatment observations, the weights are not identified. Synthetic control requires a reasonably long pre-treatment panel.
4. Skipping the permutation test. Reporting a large post-treatment gap without conducting placebo tests does not establish statistical significance.

8 Extensions and Where to Learn More

The original synthetic control has been extended in several directions:

• Augmented synthetic control (ASCM): Ben-Michael et al. [2021] augment the synthetic control estimator with an outcome model that corrects for pre-treatment imbalance, analogous to augmented IPW in the matching literature.
• Synthetic DiD: Arkhangelsky et al. [2021] combine DiD and synthetic control by using both unit weights (as in SC) and time weights (to upweight pre-treatment periods similar to the post-treatment period).
• Multiple treated units: Abadie et al. [2015] and related work extend the framework to settings with several treated units.

For implementation, the Synth package in R [Abadie et al., 2011] provides the original Abadie-Diamond-Hainmueller estimator, while augsynth and synthdid implement the extensions.

9 Conclusion

The synthetic control method addresses a fundamental problem in causal inference: how to construct a credible counterfactual for a single treated unit. By building the counterfactual as a weighted combination of control units that closely tracks the treated unit in the pre-treatment period, it makes the construction of the comparison group transparent and disciplined. Good pre-treatment fit, permutation-based inference, and a substantively chosen donor pool are the three pillars of a credible synthetic control analysis.

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. Abadie, A., Diamond, A., and Hainmueller, J. (2011). Synth: An R package for synthetic control methods in comparative case studies. Journal of Statistical Software, 42(13):1-17.
3. Abadie, A., Diamond, A., and Hainmueller, J. (2015). Comparative politics and the synthetic control method. American Journal of Political Science, 59(2):495-510.
4. Abadie, A. (2021). Using synthetic controls: Feasibility, data requirements, and methodological aspects. Journal of Economic Literature, 59(2):391-425.
5. 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.
6. Ben-Michael, E., Feller, A., and Rothstein, J. (2021). The augmented synthetic control method. Journal of the American Statistical Association, 116(536):1789-1803.