Synthetic control method: Difference between revisions

Revision as of 16:15, 1 October 2015

Revision as of 16:16, 1 October 2015

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==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>==

==Synthetic Control Method models==

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 invention with respect to the outcome 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>

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<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 pre-intervention period.

<ref>\Synthetic Control Methods for Comparative Case Studies: Estimating the Effect of California's Tobacco Control Program."</ref>