Synthetic control method: Difference between revisions

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<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''<math>w_2, \ldots, w_J</math> such that <math>Y_{1t} = \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>{{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>

Synthetic controls have been used in a number of empirical applications, ranging from studies examining natural catastrophies and growth,<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 }}</ref> and studies linking political murders to house prices.<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>