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. This comparison is used to estimate what would have happened the treatment group if it had not received the treatment.

Abadie et al. (2010) motivate the '''synthetic control method''' with a model that generalizes the [[difference-in-differences]] (fixed-effects) model commonly applied in the empirical social science literature by allowing the effect of unobserved confounding characteristics to vary over time. An attractive feature of the synthetic control method is that it guards against extrapolation outside the [[convex hull]] of the data because weights from all control units can be chosen to be positive and sum to one.

Unlike [[differences-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}}</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|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 [[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}}</ref> [[politics]],<ref name=ajps/> 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}}</ref>

==Synthetic Control Method models<ref>\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.</ref>==

To construct our synthetic control unit, the vector of weights <math>W=(w_2,w_3,...,w_{J+1})'</math> such that <math> w_j</math> ≥ O, for j=2,...,J+1 and <math>w_2+w_3+...+w_{J+1}=1</math>. Each W represents one particular weighted average of control units and therefore one potential synthetic control unit. The goal is to optimize the W* such that the resulting synthetic control unit best approximates the unit exposed to the intervention with respect to the outcome predictors <math>U_i</math> and <math>M</math> linear combinations of pre-intervention outcomes <math>\bar{Y_i}^{K_1},...,\bar{Y_i}^{K_M}</math> where <math>W^*=w_2^*+...+w_{J+1}^*</math>

such that:

<math>\sum_{j=2}^{J+1} w_j^*\bar{Y_1}^{K_1} = \bar{Y_1}^{K_1},</math>,

...,

<math>\sum_{j=2}^{J+1} w_j^*\bar{Y_1}^{K_M} = \bar{Y_1}^{K_M},</math> and <math>\sum_{j=2}^{J+1} w_j^*U_j = U_1</math> hold.

Then : <math>\hat\alpha_{1t}= Y_{1t} - \sum_{j=2}^{J+1} w_j^*Y_{jt} </math>

yields an estimator of <math>\alpha_{1t}</math> in periods <math>T_0+1,T_0+2,...,T</math>

The W* was solved by minimizing:

<math> \left\| \mathbf{X_1-X_0W} \right\|_V = \sqrt{(X_1-X_0W)'V(X_1-X_0W)}</math>, where ''''V'''' is defined as (k×k) symmetric and positive semidefinite matrix. V* is chosen among all positive definite and diagnal matrices such that the mean square prediction error (MSPE) of the outcome variable is minimized over the pre-intervention period.

==References==

{{Reflist}}