Synthetic control method: Difference between revisions

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The '''synthetic control method''' is a statistical method used to evaluate the effect of an intervention in [[comparative case study|comparative case studies]]. It involves the construction of a weighted combination of groups used as controls, to which the [[treatment group]] is compared. This comparison is used to estimate what would have happened to the treatment group if it had not received the treatment.

Unlike [[difference in differences]] approaches, this method can account for the effects of [[confounder]]s changing over time, by weighting the control group to better match the treatment group before the intervention.<ref name=he>{{cite journal|last1=Kreif|first1=Noémi|last2=Grieve|first2=Richard|last3=Hangartner|first3=Dominik|last4=Turner|first4=Alex James|last5=Nikolova|first5=Silviya|last6=Sutton|first6=Matt|title=Examination of the Synthetic Control Method for Evaluating Health Policies with Multiple Treated Units|journal=Health Economics|date=December 2016|volume=25|issue=12|pages=1514–1528|doi=10.1002/hec.3258}}</ref> Another advantage of the synthetic control method is that it allows researchers to systematically select comparison groups.<ref name=ajps>{{cite journal|last1=Abadie|first1=Alberto|last2=Diamond|first2=Alexis|last3=Hainmueller|first3=Jens|title=Comparative Politics and the Synthetic Control Method|journal=American Journal of Political Science|date=February 2015|volume=59|issue=2|pages=495–510|doi=10.1111/ajps.12116}}</ref> It has been applied to the fields of [[health policy]],<ref name=he/> [[criminology]],<ref>{{cite journal|last1=Saunders|first1=Jessica|last2=Lundberg|first2=Russell|last3=Braga|first3=Anthony A.|last4=Ridgeway|first4=Greg|last5=Miles|first5=Jeremy|title=A Synthetic Control Approach to Evaluating Place-Based Crime Interventions|journal=Journal of Quantitative Criminology|date=3 June 2014|volume=31|issue=3|pages=413–434|doi=10.1007/s10940-014-9226-5}}</ref> [[politics]],<ref name=ajps/> and [[economics]].<ref>{{cite journal|last1=Billmeier|first1=Andreas|last2=Nannicini|first2=Tommaso|title=Assessing Economic Liberalization Episodes: A Synthetic Control Approach|journal=Review of Economics and Statistics|date=July 2013|volume=95|issue=3|pages=983–1001|doi=10.1162/REST_a_00324}}</ref>

The synthetic control method combines elements from [[Matching (statistics)|matching]] and [[difference-in-difference]] techniques. 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,<ref name="CardKrueger">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.</ref> and studies that look at crime rates in southern cities to evaluate the impact of the Mariel boat lift on crime.<ref>Card, D. (1990): The Impact of the Mariel Boatlift on the Miami Labor Market. Industrial and Labor Relations Review 44, pp. 245-257.</ref> 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

<math>T_{0}</math> where <math>T_{0}<T.</math> Let <math>\alpha_{it}=Y_{it}-Y^N_{it},</math> where <math>Y^N_{it}</math> 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 <math>Y^N_{1t}</math>is not observed for <math>t>T_{0}</math> and we aim to estimate (<math>(\alpha_{1T_{0+1}}......\alpha_{1T})</math>. Imposing some structure

<math>Y^N_{it}=\delta_{t}+\theta_{t}Z_{i}+\lambda_{t}\mu_{i}+\varepsilon_{it}</math> and assuming there exist some optimal weights ''w'' such that <math>\Sigma^J_{j=2} w_{j}Y_{jt}</math> for <math>t\leqslant T_{0}</math>, to the synthetic controls approach suggests using these weights to estimate the counterfactual <math>Y^N_{1t}=\Sigma^J_{j=2}w_{j}Y_{jt}</math> for <math>t>T_{0}</math>. 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.<ref>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.</ref>

Synthetic controls have been used in a number of empirical applications, ranging from studies examining natural catastrophies and growth,<ref>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.</ref> and studies linking political murders to house prices.<ref>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.</ref> Yet, despite its intuitive appeal, it may be the case that synthetic controls could suffer from significant finite sample biases.<ref>Devereux, P. J. (2007): Small-sample bias in synthetic cohort models of labor supply. Journal of Applied Econometrics 22(4), pp. 839-848.</ref>

==References==

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[[Category:Design of experiments]]

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