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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.

'''合成对照 synthetic control method'''是一种统计方法，在比较案例研究中用于评估干预措施的效果。它使用多组数据的加权组合来构建对照组，使之与治疗组进行比较。<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>基于这种比较，可用来估计在干预之后的时间里，假如没有对治疗组进行干预的情况下治疗组将如何发展。

【翻译】合成对照是一种统计方法，在比较案例研究中用于评估干预措施的效果。它使用多组数据的加权组合来构建对照组，使之与治疗组进行比较。基于这种比较，可用来估计在干预之后的时间里，假如没有对治疗组进行干预的情况下治疗组将如何发展。与双重差分方法（Difference in difference）不同，这种方法可以考虑混杂因素随时间变化的影响，通过调整对照组的加权组合，可以对干预之前的治疗组数据做更好的匹配。合成对照还有个优点是，它允许研究人员在多组候选数据中做系统性选择。它已应用于政治学、卫生政策、犯罪学和经济学等多个领域。

与双重差分方法 Difference in difference不同，这种方法可以考虑混杂因素随时间变化的影响，通过调整对照组的加权组合，可以对干预之前的治疗组数据做更好的匹配。<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>合成对照还有个优点是，它允许研究人员在多组候选数据中做系统性选择。<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>它已应用于政治学<ref name=ajps/>、卫生政策<ref name=he/>、犯罪学<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>和经济学等多个领域。<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 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, and studies that look at crime rates in southern cities to evaluate the impact of the Mariel boat lift on crime. 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.

合成对照方法结合了匹配方法和双重差分方法的技术要素。双重差分法也是一种常用的政策评估工具，通过比较被干预单元和未被干预单元在总体水平上（例如：州、国家、年龄组别等）的均值差异来评估政策干预效果。著名的案例包括新泽西州快餐店提高最低工资政策对就业影响的研究，<ref name="CardKrueger">{{cite journal |last=Card |first=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>比较对象是在新泽西州边界另一侧，费城那边那些没受到该政策影响的快餐店；还有通过研究南部城市的犯罪率来评估马里埃尔移民潮如何影响犯罪的案例。<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 |url=http://arks.princeton.edu/ark:/88435/dsp016h440s46f }}</ref>在双重差分场景中，合成对照的控制组可被理解为一个加权平均，其中的一些单元相当于得到了零权重，而另外的一些单元则得到了非零权重（每个单元内的数据共享同一权重值）。

合成对照方法试图用一种更加系统的方法为控制组分配权重。它通常用干预之前比较长一段时间内的多个时间序列作为输入数据，估计一组权重值使得这些输入数据加权的结果尽可能地拟合治疗组的时间序列数据，并将结果用作控制组时间序列数据。特别地，假设我们在T个时间段里共有J个观测量（单元），其中一个单元在<math>T_{0}</math>时间接受了治疗，<math>T_{0}<T.</math>。让

 

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>

:<math>\alpha_{it}=Y_{it}-Y^N_{it},</math>

为单元<math>i</math>的在时间<math>t</math>的治疗效果，其中<math>Y^N_{it}</math>是未经治疗的结果。不失一般性，如果指定单元1接受治疗，则只有单元1的数据<math>Y^N_{1t}</math>在 <math>t>T_{0}</math>时段是无法观测的。而我们的目标是要估计<math>(\alpha_{1T_{0}+1}......\alpha_{1T})</math>的值。

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

: <math>Y^N_{it}=\delta_{t}+\theta_{t}Z_{i}+\lambda_{t}\mu_{i}+\varepsilon_{it}</math>

 

假设存在一些最优权重<math>w_2, \ldots, w_J</math>，使得

 

 

 

 

 

 

 

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

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

: <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 name=":0">{{cite journal |last=Abadie |first=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>

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

因此，在一定的正则性条件下，此类权重可以作为我们所关心的治疗效果的估计量。本质上，该方法基于匹配的思想，利用干预前的数据训练得到加权组合的控制组，进而可以对干预后的控制组数据进行推断。<ref name=":0">{{cite journal |last=Abadie |first=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>

and assuming there exist some optimal weights w_2, \ldots, w_J such that

合成对照已经被应用于许多实证研究中，从研究自然灾害和经济增长，<ref>{{cite journal |last=Cavallo |first=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  |url=http://www.economics.hawaii.edu/research/workingpapers/WP_10-6.pdf }}</ref>到研究政治谋杀与房价之间的联系。<ref>{{cite journal |last=Gautier |first=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 }}</ref>

:Y^N_{1t}=\Sigma^J_{j=2}w_{j}Y_{jt}

==参考文献==

{{Reflist|30em}}

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