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.<ref>{{Cite journal|last=Abadie|first=Alberto|date=2021|title=Using Synthetic Controls: Feasibility, Data Requirements, and Methodological Aspects|url=https://www.aeaweb.org/articles?id=10.1257/jel.20191450|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}}</ref> 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|pmid=26443693|pmc=5111584}}</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|authorlink1=Alberto Abadie|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 [[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 [[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|s2cid=57561957 }}</ref>

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 |lastlast1=Card |firstfirst1=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]] on crime.<ref>{{cite journal |last=Card |first=D. |year=1990 |title=The Impact of the Mariel Boatlift on the Miami Labor Market |journal=[[Industrial and Labor Relations Review]] |volume=43 |issue=2 |pages=245–257 |doi=10.1177/001979399004300205 |s2cid=15116852 |url=http://arks.princeton.edu/ark:/88435/dsp016h440s46f }}</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

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:<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>{{cite journal |lastlast1=Abadie |firstfirst1=A. |authorlink=Alberto Abadie |first2=A. |last2=Diamond |first3= J. |last3=Hainmüller |year=2010 |title=Synthetic Control Methods for Comparative Case Studies: Estimating the Effect of California's Tobacco Control Program |journal=[[Journal of the American Statistical Association]] |volume=105 |issue=490 |pages=493–505 |doi=10.1198/jasa.2009.ap08746 }}</ref>

Synthetic controls have been used in a number of empirical applications, ranging from studies examining natural catastrophes and growth,<ref>{{cite journal |lastlast1=Cavallo |firstfirst1=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 immunisation,<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 |lastlast1=Gautier |firstfirst1=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> -->