Synthetic controls

Synthetic Controls

Synthetic controls is a relatively new econometric technique that combines elements from the matching and difference-in-difference literature and aims at providing a relevant control group. It was proposed by Abadie, Diamond, and Hainmuller [1].2]

Difference-in-difference 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 [2] , and studies that look at crime rates in southern cities to evaluate the impact of the Mariel boat lift on crime [3]. 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

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

$Y_{it}^{N}=\delta _{t}+\theta _{t}Z_{i}+\lambda _{t}\mu _{i}+\varepsilon _{it}$ and assuming there exist some optimal weights w such that $\Sigma _{j=2}^{J}w_{j}Y_{jt}$ for $t\leqslant T_{0}$, to the synthetic controls approach suggests using these weights to estimate the counterfactual $Y_{1t}^{N}=\Sigma _{j=2}^{J}w_{j}Y_{jt}$ for $t>T_{0}$. 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 [4].

Synthetic controls have been used in a number of empirical applications, ranging from studies examining natural catastrophies and growth [5] and studies linking political murders to house prices [6]. Yet, despite its intuitive appeal, it may be the case that synthetic controls could suffer from significant finite sample biases[7].

References

1. Abadie, A., A. Diamond, and J. Hainmuller (2010): Synthetic Control Methods for Comparative Case Studies: Estimating the Effect of California's Tobacco Control Program. Journal of the American Statistical Association 105, pp. 493-505.
2. Card, D. and A. Krueger (1994): Minimum Wages and Employment: A Case Study of the Fast-Food Industry in New Jersey and Pennsylvania. American Economic Revie 84, pp. 772-793.
3. Card, D. (1990): The Impact of the Mariel Boatlift on the Miami Labor Market. Industrial and Labor Relations Review 44, pp. 245-257.
4. Abadie, A., A. Diamond, and J. Hainmuller (2010): Synthetic Control Methods for Comparative Case Studies: Estimating the Effect of California's Tobacco Control Program. Journal of the American Statistical Association 105, pp. 493-505.
5. Cavallo, E., S. Galliani, I. Noy, and J. Pantano (2013): Catastropic Natural Disasters and Economic Growth. Review of Economics and Statistics 95(5), pp. 1549-1561.
6. Gautier, P. A., A. Siegmann, and A. Van Vuuren (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), pp. 113-126.
7. Devereux, P. J. (2007): Small-sample bias in synthetic cohort models of labor supply. Journal of Applied Econometrics 22(4), pp. 839-848.

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