Synthetic control method

The synthetic control method is an econometric method used to evaluate the effect of large-scale interventions. It was proposed in a series of articles by Alberto Abadie and his coauthors.^[2][3][4] A synthetic control is a weighted average of several units (such as regions or companies) combined to recreate the trajectory that the outcome of a treated unit would have followed in the absence of the intervention. The weights are selected in a data-driven manner to ensure that the resulting synthetic control closely resembles the treated unit in terms of key predictors of the outcome variable.^[2] Unlike difference in differences approaches, this method can account for the effects of confounders changing over time, by weighting the control group to better match the treatment group before the intervention.^[5] Another advantage of the synthetic control method is that it allows researchers to systematically select comparison groups. It has been applied to the fields of economics,^[6] political science,^[1] health policy,^[5] criminology,^[7] and others.

Comparison of per-capita GDP in West Germany before and after the 1990 German reunification and the hypothetical one if the reunification had not taken place.^[1]

The synthetic control method combines elements from matching and difference-in-differences techniques. Difference-in-differences methods are often-used policy evaluation tools that estimate the effect of an intervention at an aggregate level (e.g. state, country, age group etc.) by averaging over a set of unaffected units. Famous examples include studies of the employment effects of a raise in the minimum wage in New Jersey fast food restaurants by comparing them to fast food restaurants just across the border in Philadelphia that were unaffected by a minimum wage raise,^[8] and studies that look at crime rates in southern cities to evaluate the impact of the Mariel Boatlift on crime.^[9] The control group in this specific scenario can be interpreted as a weighted average, where some units effectively receive zero weight while others get an equal, non-zero weight.

The synthetic control method tries to offer a more systematic way to assign weights to the control group. It typically uses a relatively long time series of the outcome prior to the intervention and estimates weights in such a way that the control group mirrors the treatment group as closely as possible. In particular, assume we have J observations over T time periods where the relevant treatment occurs at time ${\displaystyle T_{0}}$ where ${\displaystyle T_{0}<T.}$ Let

${\displaystyle \alpha _{it}=Y_{it}-Y_{it}^{N},}$

be the treatment effect for unit ${\displaystyle i}$ at time ${\displaystyle t}$, where ${\displaystyle Y_{it}^{N}}$ is the outcome in absence of the treatment. Without loss of generality, if unit 1 receives the relevant treatment, only ${\displaystyle Y_{1t}^{N}}$ is not observed for ${\displaystyle t>T_{0}}$. We aim to estimate ${\displaystyle (\alpha _{1T_{0}+1}......\alpha _{1T})}$.

Imposing some structure

${\displaystyle Y_{it}^{N}=\delta _{t}+\theta _{t}Z_{i}+\lambda _{t}\mu _{i}+\varepsilon _{it}}$

and assuming there exist some optimal weights ${\displaystyle w_{2},\ldots ,w_{J}}$ such that

${\displaystyle Y_{1t}=\sum _{j=2}^{J}w_{j}Y_{jt}}$

for ${\displaystyle t\leqslant T_{0}}$, the synthetic controls approach suggests using these weights to estimate the counterfactual

${\displaystyle Y_{1t}^{N}=\sum _{j=2}^{J}w_{j}Y_{jt}}$

for ${\displaystyle t>T_{0}}$. So under some regularity conditions, such weights would provide estimators for the treatment effects of interest. In essence, the method uses the idea of matching and using the training data pre-intervention to set up the weights and hence a relevant control post-intervention.^[2]

Synthetic controls have been used in a number of empirical applications, ranging from studies examining natural catastrophes and growth,^[10] studies that examine the effect of vaccine mandates on childhood immunization,^[11] and studies linking political murders to house prices.^[12]

See also

• Difference in difference
• Regression discontinuity
• Instrumental variables estimation

References

1. Abadie, Alberto; Diamond, Alexis; Hainmueller, Jens (February 2015). "Comparative Politics and the Synthetic Control Method". American Journal of Political Science. 59 (2): 495–510. doi:10.1111/ajps.12116.
2. Abadie, Alberto; Gardeazabal, Javier (2003). "The Economic Costs of Conflict: A Case Study of the Basque Country". American Economic Review. 93 (1): 113–132. doi:10.1257/000282803321455188. ISSN 0002-8282.
3. Abadie, Alberto; Diamond, Alexis; Hainmueller, Jens (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. doi:10.1198/jasa.2009.ap08746. ISSN 0162-1459.
4. Abadie, Alberto (2021). "Using Synthetic Controls: Feasibility, Data Requirements, and Methodological Aspects". Journal of Economic Literature. 59 (2): 391–425. doi:10.1257/jel.20191450. hdl:1721.1/144417. ISSN 0022-0515.
5. Kreif, Noémi; Grieve, Richard; Hangartner, Dominik; Turner, Alex James; Nikolova, Silviya; Sutton, Matt (December 2016). "Examination of the Synthetic Control Method for Evaluating Health Policies with Multiple Treated Units". Health Economics. 25 (12): 1514–1528. doi:10.1002/hec.3258. PMC 5111584. PMID 26443693.
6. Billmeier, Andreas; Nannicini, Tommaso (July 2013). "Assessing Economic Liberalization Episodes: A Synthetic Control Approach". Review of Economics and Statistics. 95 (3): 983–1001. doi:10.1162/REST_a_00324. S2CID 57561957.
7. Saunders, Jessica; Lundberg, Russell; Braga, Anthony A.; Ridgeway, Greg; Miles, Jeremy (3 June 2014). "A Synthetic Control Approach to Evaluating Place-Based Crime Interventions". Journal of Quantitative Criminology. 31 (3): 413–434. doi:10.1007/s10940-014-9226-5. S2CID 254702864.
8. Card, D.; Krueger, A. (1994). "Minimum Wages and Employment: A Case Study of the Fast-Food Industry in New Jersey and Pennsylvania". American Economic Review. 84 (4): 772–793. JSTOR 2118030.
9. Billy, Alexander (2022). "Crime and the Mariel Boatlift". International Review of Law and Economics. 72: 106094. doi:10.1016/j.irle.2022.106094. S2CID 219390309 – via Science Direct.
10. Cavallo, E.; Galliani, S.; Noy, I.; Pantano, J. (2013). "Catastrophic Natural Disasters and Economic Growth" (PDF). Review of Economics and Statistics. 95 (5): 1549–1561. doi:10.1162/REST_a_00413. S2CID 16038784.
11. Li, Ang.; Toll, Mathew. (2020). "Removing conscientious objection: The impact of 'No Jab No Pay' and 'No Jab No Play' vaccine policies in Australia". Preventive Medicine. 145: 106406. doi:10.1016/j.ypmed.2020.106406. ISSN 0091-7435. PMID 33388333. S2CID 230489130.
12. Gautier, P. A.; Siegmann, A.; Van Vuuren, A. (2009). "Terrorism and Attitudes towards Minorities: The effect of the Theo van Gogh murder on house prices in Amsterdam". Journal of Urban Economics. 65 (2): 113–126. doi:10.1016/j.jue.2008.10.004. S2CID 190624.