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The synthetic control method is a statistical method used to evaluate the effect of an intervention in comparative case studies. It involves the construction of a weighted combination of groups used as controls, to which the treatment group is compared.[1] 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 confounders changing over time, by weighting the control group to better match the treatment group before the intervention.[2] Another advantage of the synthetic control method is that it allows researchers to systematically select comparison groups.[3] It has been applied to the fields of political science,[3] health policy,[2] criminology,[4] and economics.[5]
The synthetic control method is a statistical method used to evaluate the effect of an intervention in 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 confounders changing over time, by weighting the control group to better match the treatment group before the intervention. 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 political science, health policy, criminology, and economics.
综合控制方法是一种统计方法，用于评估比较案例研究中的干预措施的效果。它包括用作对照组的加权组合的构建，并与治疗组进行比较。这种比较被用来估计如果治疗组没有接受治疗会发生什么。与不同方法的差异不同，这种方法可以解释随时间变化的混杂因素的影响，通过加权对照组，以便在干预前更好地匹配治疗组。综合控制法的另一个优点是，它允许研究人员系统地选择对照组。它已应用于政治学、卫生政策、犯罪学和经济学等领域。
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,[6] and studies that look at crime rates in southern cities to evaluate the impact of the Mariel boat lift on crime.[7] 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 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.
综合控制方法结合了匹配技术和差中差技术的要素。差异中的差异法是一种常用的政策评估工具，用于在总体水平上评估干预措施的效果(例如:。州、国家、年龄组别等)平均超过一组未受影响的单位。著名的例子包括新泽西州快餐店提高最低工资对就业影响的研究，通过比较它们与费城边境那边没有受到最低工资提高影响的快餐店，以及研究南部城市的犯罪率来评估马里埃尔船只提升对犯罪的影响。在这个特定的场景中，控制组可以被解释为一个加权平均数，其中一些单位实际上得到了零重量，而其他单位得到了相等的，非零重量。
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]\displaystyle{ T_{0} }[/math] where [math]\displaystyle{ T_{0}\lt T. }[/math] Let
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
综合控制方法试图为控制组的权重分配提供一种更加系统的方法。它通常使用一个相对较长的时间序列的结果之前的干预和估计权重的方式，控制组镜像治疗组尽可能接近。特别地，假设我们在 t 时间段有 j 观测值，在 t {0} < t 时相应的处理发生在 t {0} < t 时。让
- [math]\displaystyle{ \alpha_{it}=Y_{it}-Y^N_{it}, }[/math]
be the treatment effect for unit [math]\displaystyle{ i }[/math] at time [math]\displaystyle{ t }[/math], where [math]\displaystyle{ 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]\displaystyle{ Y^N_{1t} }[/math]is not observed for [math]\displaystyle{ t\gt T_{0} }[/math]. We aim to estimate [math]\displaystyle{ (\alpha_{1T_{0}+1}......\alpha_{1T}) }[/math].
- \alpha_{it}=Y_{it}-Y^N_{it},
be the treatment effect for unit i at time t, where Y^N_{it} is the outcome in absence of the treatment. Without loss of generality, if unit 1 receives the relevant treatment, only Y^N_{1t}is not observed for t>T_{0}. We aim to estimate (\alpha_{1T_{0}+1}......\alpha_{1T}).
- alpha _ { it } = y _ { it }-y ^ n _ { it } ，为时间 t 的单位 i 的治疗效果，其中 y ^ n _ { it }是未经治疗的结果。不失一般性，如果单位1接受相应的治疗，只有 y ^ n {1 t }没有观察到 t > t {0}。我们的目标是估计(alpha _ {1T _ {0} + 1} ... ... alpha _ {1T })。
Imposing some structure
Imposing some structure
强加一些结构
- [math]\displaystyle{ Y^N_{it}=\delta_{t}+\theta_{t}Z_{i}+\lambda_{t}\mu_{i}+\varepsilon_{it} }[/math]
- Y^N_{it}=\delta_{t}+\theta_{t}Z_{i}+\lambda_{t}\mu_{i}+\varepsilon_{it}
- y ^ n { it } = delta { t } + theta { t } z { i } + lambda { t } mu { i } + varepsilon { it }
and assuming there exist some optimal weights [math]\displaystyle{ w_2, \ldots, w_J }[/math] such that
and assuming there exist some optimal weights w_2, \ldots, w_J such that
假设存在一些最优权重 w _ 2，ldots，w _ j 使得
- [math]\displaystyle{ Y_{1t} = \Sigma^J_{j=2} w_{j}Y_{jt} }[/math]
- Y_{1t} = \Sigma^J_{j=2} w_{j}Y_{jt}
- y {1 t } = Sigma ^ j { j = 2} w { j } y { jt }
for [math]\displaystyle{ t\leqslant T_{0} }[/math], the synthetic controls approach suggests using these weights to estimate the counterfactual
for t\leqslant T_{0}, the synthetic controls approach suggests using these weights to estimate the counterfactual
对于 t _ {0} ，综合控制方法建议使用这些权重来估计反事实
- [math]\displaystyle{ Y^N_{1t}=\Sigma^J_{j=2}w_{j}Y_{jt} }[/math]
for [math]\displaystyle{ t\gt 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.[8]
- Y^N_{1t}=\Sigma^J_{j=2}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.
- Y^N_{1t}=\Sigma^J_{j=2}w_{j}Y_{jt}
for t>T_{0}.因此，在一定的正则性条件下，这种权重可以作为利息处理效果的估计量。本质上，该方法采用了匹配的思想，并利用训练数据进行干预前的权重设置，从而得到干预后的相应控制。
Synthetic controls have been used in a number of empirical applications, ranging from studies examining natural catastrophes and growth,[9] and studies linking political murders to house prices.[10]
Synthetic controls have been used in a number of empirical applications, ranging from studies examining natural catastrophes and growth, and studies linking political murders to house prices.
综合控制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，到研究政治谋杀与房价之间的联系。
References
- ↑ Abadie, Alberto (2021). "Using Synthetic Controls: Feasibility, Data Requirements, and Methodological Aspects". Journal of Economic Literature (in English). 59 (2): 391–425. doi:10.1257/jel.20191450. ISSN 0022-0515.
- ↑ 2.0 2.1 Kreif, Noémi; Grieve, Richard; Hangartner, Dominik; Turner, Alex James; Nikolova, Silviya; Sutton, Matt (December 2016). "Examination of the Synthetic Control Method for Evaluating Health Policies with Multiple Treated Units". Health Economics. 25 (12): 1514–1528. doi:10.1002/hec.3258. PMC 5111584. PMID 26443693.
- ↑ 3.0 3.1 Abadie, Alberto; Diamond, Alexis; Hainmueller, Jens (February 2015). "Comparative Politics and the Synthetic Control Method". American Journal of Political Science. 59 (2): 495–510. doi:10.1111/ajps.12116.
- ↑ Saunders, Jessica; Lundberg, Russell; Braga, Anthony A.; Ridgeway, Greg; Miles, Jeremy (3 June 2014). "A Synthetic Control Approach to Evaluating Place-Based Crime Interventions". Journal of Quantitative Criminology. 31 (3): 413–434. doi:10.1007/s10940-014-9226-5.
- ↑ Billmeier, Andreas; Nannicini, Tommaso (July 2013). "Assessing Economic Liberalization Episodes: A Synthetic Control Approach". Review of Economics and Statistics. 95 (3): 983–1001. doi:10.1162/REST_a_00324.
- ↑ Card, D.; Krueger, A. (1994). "Minimum Wages and Employment: A Case Study of the Fast-Food Industry in New Jersey and Pennsylvania". American Economic Review. 84 (4): 772–793. JSTOR 2118030.
- ↑ Card, D. (1990). "The Impact of the Mariel Boatlift on the Miami Labor Market". Industrial and Labor Relations Review. 43 (2): 245–257. doi:10.1177/001979399004300205.
- ↑ Abadie, A.; Diamond, A.; Hainmüller, J. (2010). "Synthetic Control Methods for Comparative Case Studies: Estimating the Effect of California's Tobacco Control Program". Journal of the American Statistical Association. 105 (490): 493–505. doi:10.1198/jasa.2009.ap08746.
- ↑ Cavallo, E.; Galliani, S.; Noy, I.; Pantano, J. (2013). "Catastrophic Natural Disasters and Economic Growth" (PDF). Review of Economics and Statistics. 95 (5): 1549–1561. doi:10.1162/REST_a_00413.
- ↑ Gautier, P. A.; Siegmann, A.; Van Vuuren, A. (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): 113–126. doi:10.1016/j.jue.2008.10.004.
Category:Design of experiments Category:Statistical methods Category:Observational study Category:Econometric modeling
类别: 实验设计类别: 统计方法类别: 观察性研究类别: 计量经济模型
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