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Synthetic control method
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Synthetic control method : ウィキペディア英語版
Synthetic control method
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.
==Synthetic Control Method models〔\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=1. 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 U_i and M linear combinations of pre-intervention outcomes \bar^,...,\bar^ where W^
*=w_2^
*+...+w_^
*
such that:
\sum_^ w_j^
*\bar^ = \bar^,,
...,
\sum_^ w_j^
*\bar^ = \bar^, and \sum_^ w_j^
*U_j = U_1 hold.
Then : \hat\alpha_= Y_1t - \sum_^ w_j^
*Y_
yields an estimator of \alpha_ in periods T_0+1,T_0+2,...,T
The W
* was solved by minimize:
\left\| \mathbf \right\|_V = \sqrt, 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.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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