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

To construct our synthetic control unit, the vector of weights $W=(w_{2},w_{3},...,w_{J+1})'$ such that $w_{j}$ ≥ O, for j=2,...,J+1 and $w_{2}+w_{3}+...+w_{J+1}=1$. 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 ${\bar {Y_{i}}}^{K_{1}},...,{\bar {Y_{i}}}^{K_{M}}$ where $W^{*}=w_{2}^{*}+...+w_{J+1}^{*}$

such that: $\sum _{j=2}^{J+1}w_{j}^{*}{\bar {Y_{1}}}^{K_{1}}={\bar {Y_{1}}}^{K_{1}},$, ..., $\sum _{j=2}^{J+1}w_{j}^{*}{\bar {Y_{1}}}^{K_{M}}={\bar {Y_{1}}}^{K_{M}},$ and $\sum _{j=2}^{J+1}w_{j}^{*}U_{j}=U_{1}$ hold.

Then : ${\hat {\alpha }}_{1t}=Y_{1}t-\sum _{j=2}^{J+1}w_{j}^{*}Y_{jt}$

yields an estimator of $\alpha _{1t}$ in periods $T_{0}+1,T_{0}+2,...,T$

The W* was solved by minimize:

$\left\|\mathbf {X_{1}-X_{0}W} \right\|_{V}={\sqrt {(X_{1}-X_{0}W)'V(X_{1}-X_{0}W)}}$, 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.