Synthetic control method: Difference between revisions

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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 |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> 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 |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

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

<math>T_{0}</math>be wherethe treatment effect for unit <math>T_{0}<T.i</math> Letat time <math>\alpha_{it}=Y_{it}-Y^N_{it},t</math>, where <math>Y^N_{it}</math> is 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 weWe aim to estimate <math>(\alpha_{1T_{0}+1}......\alpha_{1T})</math>. Imposing some structure

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} = \Sigma^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}=\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>

:<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>

for <math>t\leqslant T_{0}</math>, 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>

<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> for <math>t\leqslant T_{0}</math>, 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 catastrophes 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 |url=http://www.economics.hawaii.edu/research/workingpapers/WP_10-6.pdf }}</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>