View source for Synthetic control method

{{Short description|Type of statistical data method}}
[[File:SCMGermany.png|thumb|right|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.<ref name="ajps">{{cite journal |last1=Abadie |first1=Alberto |last2=Diamond |first2=Alexis |last3=Hainmueller |first3=Jens |date=February 2015 |title=Comparative Politics and the Synthetic Control Method |journal=American Journal of Political Science |volume=59 |issue=2 |pages=495–510 |doi=10.1111/ajps.12116 |authorlink1=Alberto Abadie}}</ref>|260x260px]]
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.<ref name=":0">{{Cite journal |last=Abadie |first=Alberto |last2=Gardeazabal |first2=Javier |date=2003 |title=The Economic Costs of Conflict: A Case Study of the Basque Country |url=https://pubs.aeaweb.org/doi/10.1257/000282803321455188 |journal=American Economic Review |language=en |volume=93 |issue=1 |pages=113–132 |doi=10.1257/000282803321455188 |issn=0002-8282}}</ref><ref>{{Cite journal |last=Abadie |first=Alberto |last2=Diamond |first2=Alexis |last3=Hainmueller |first3=Jens |date=2010 |title=Synthetic Control Methods for Comparative Case Studies: Estimating the Effect of California’s Tobacco Control Program |url=http://www.tandfonline.com/doi/abs/10.1198/jasa.2009.ap08746 |journal=Journal of the American Statistical Association |language=en |volume=105 |issue=490 |pages=493–505 |doi=10.1198/jasa.2009.ap08746 |issn=0162-1459}}</ref><ref name=":1">{{Cite journal |last=Abadie |first=Alberto |date=2021 |title=Using Synthetic Controls: Feasibility, Data Requirements, and Methodological Aspects |journal=Journal of Economic Literature |language=en |volume=59 |issue=2 |pages=391–425 |doi=10.1257/jel.20191450 |issn=0022-0515 |doi-access=free |hdl-access=free |hdl=1721.1/144417}}</ref> 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.<ref name=":0" /> 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|pmid=26443693|pmc=5111584}}</ref> 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]],<ref>{{cite journal |last1=Billmeier |first1=Andreas |last2=Nannicini |first2=Tommaso |date=July 2013 |title=Assessing Economic Liberalization Episodes: A Synthetic Control Approach |journal=Review of Economics and Statistics |volume=95 |issue=3 |pages=983–1001 |doi=10.1162/REST_a_00324 |s2cid=57561957}}</ref> [[political science]],<ref name="ajps" /> [[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|s2cid=254702864 }}</ref> and others.

The synthetic control method combines elements from [[Matching (statistics)|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 the United States|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">{{cite journal |last1=Card |first1=D. |authorlink=David Card |first2=A. |last2=Krueger |authorlink2=Alan Krueger |year=1994 |title=Minimum Wages and Employment: A Case Study of the Fast-Food Industry in New Jersey and Pennsylvania |journal=[[American Economic Review]] |volume=84 |issue=4 |pages=772–793 |jstor=2118030 }}</ref> and studies that look at [[crime rates]] in southern cities to evaluate the impact of the [[Mariel boat lift|Mariel Boatlift]] on crime.<ref>{{cite journal |last=Billy |first=Alexander |year=2022 |title=Crime and the Mariel Boatlift |url=https://www.sciencedirect.com/science/article/pii/S0144818822000503 |journal=[[International Review of Law and Economics]] |volume=72 |issue= |pages=106094 |doi=10.1016/j.irle.2022.106094 |s2cid=219390309 |via=Science Direct}}</ref>  The control group in this specific scenario can be interpreted as a [[Weighted arithmetic mean|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> 
be the treatment effect for unit <math>i</math> at time <math>t</math>, where <math>Y^N_{it}</math> is the outcome in absence of the treatment. 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>. 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 <math>w_2, \ldots, w_J</math> such that

:<math>Y_{1t} = \sum^J_{j=2} w_{j}Y_{jt}</math>

for <math>t\leqslant T_{0}</math>, the synthetic controls approach suggests using these weights to estimate the counterfactual

:<math>Y^N_{1t}=\sum^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 name=":0" />

Synthetic controls have been used in a number of empirical applications, ranging from studies examining natural catastrophes and growth,<ref>{{cite journal |last1=Cavallo |first1=E. |first2=S. |last2=Galliani |first3=I. |last3=Noy |first4=J. |last4=Pantano |year=2013 |title=Catastrophic Natural Disasters and Economic Growth |journal=[[Review of Economics and Statistics]] |volume=95 |issue=5 |pages=1549–1561 |doi=10.1162/REST_a_00413  |s2cid=16038784 |url=http://www.economics.hawaii.edu/research/workingpapers/WP_10-6.pdf }}</ref> studies that examine the effect of vaccine mandates on childhood immunization,<ref>{{cite journal |last1=Li |first1=Ang. |last2=Toll |first2=Mathew. |title=Removing conscientious objection: The impact of 'No Jab No Pay' and 'No Jab No Play' vaccine policies in Australia |journal=Preventive Medicine |date=2020 |volume=145 |page=106406 |doi=10.1016/j.ypmed.2020.106406 |issn=0091-7435 |pmid=33388333 |s2cid=230489130 |url=https://www.sciencedirect.com/science/article/abs/pii/S0091743520304370}}</ref> and studies linking political murders to house prices.<ref>{{cite journal |last1=Gautier |first1=P. A. |first2=A. |last2=Siegmann |first3=A. |last3=Van Vuuren |year=2009 |title=Terrorism and Attitudes towards Minorities: The effect of the Theo van Gogh murder on house prices in Amsterdam |journal=[[Journal of Urban Economics]] |volume=65 |issue=2 |pages=113–126 |doi=10.1016/j.jue.2008.10.004 |s2cid=190624 }}</ref> 
<!--  THE CITATION AT THE END OF THIS SENTENCE IS FOR A PAPER ABOUT "synthetic cohort models" (a.k.a. "pseudo-panel approach," using repeated cross-sections), WHICH IS NOT THE SAME AS "synthetic control":  Yet, despite its intuitive appeal, it may be the case that synthetic controls could suffer from significant finite sample biases.<ref>{{cite journal |last=Devereux |first=P. J. |year=2007 |title=Small-sample bias in synthetic cohort models of labor supply |journal=[[Journal of Applied Econometrics]] |volume=22 |issue=4 |pages=839–848 |doi=10.1002/jae.938 }}</ref>  -->

== See also ==

* [[Difference in differences|Difference in difference]]
* [[Regression discontinuity design|Regression discontinuity]]
* [[Instrumental variables estimation]]

==References==
{{Reflist|30em}}

[[Category:Design of experiments]]
[[Category:Statistical methods]]
[[Category:Observational study]]
[[Category:Econometric modeling]]