Synthetic control method: Difference between revisions

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==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>


such that:

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