Synthetic-Control Formalism

Let $Y_{it}$ be the outcome for unit $i$ and period $t$. Let $\mathcal{T}$ denote treated units, $\mathcal{C}$ eligible donors, and $T_0$ the last pre-treatment period after any window resolution.

For one treated unit, a synthetic counterfactual has the form

$$ \widehat{Y}_{1t}(0) = \sum_{j \in \mathcal{C}} w_j Y_{jt}. $$

Weights and the match space are estimated from pre-treatment information. The period effect is

$$ \widehat{\tau}_t = Y_{1t} - \widehat{Y}_{1t}(0), $$

and GeoSC’s top-level unscaled ATT is the average of that effect over measured post periods:

$$ \widehat{\tau} = \frac{1}{T_1}\sum_{t=T_0+1}^{T_0+T_1}\widehat{\tau}_t. $$

The implementation also supports multiple treated units. SparseSC constructs a joint treated effect and an in-space placebo reference set formed from treated- sized subsets of controls. With $N_0$ controls and $N_1$ treated units, the exhaustive assignment count is

$$ M = \binom{N_0}{N_1}. $$

GeoSC’s top-level p-value is not a post/pre mean-squared prediction-error ratio. If $\widehat{\tau}^{(m)}$ is the average placebo effect for assignment $m$, SparseSC uses the two-sided absolute-effect comparison with add-one correction:

$$ p = \frac{1 + \sum_{m=1}^{M^*} \mathbf{1}\left\{|\widehat{\tau}^{(m)}| \geq |\widehat{\tau}|\right\}} {M^*+1}, $$

where $M^*$ is the effective exhaustive or sampled placebo count. The finest attainable step under this calculation is $1/(M^*+1)$. The result is an empirical comparison against the eligible placebo assignments; its causal meaning depends on the design.