32.7 Theoretical Considerations
SCM assumes that the counterfactual outcome follows a factor model (Abadie, Diamond, and Hainmueller 2010):
\[ Y_{it}^N = \mathbf{\theta}_t \mathbf{Z}_i + \mathbf{\lambda}_t \mathbf{\mu}_i + \epsilon_{it} \]
where:
- \(\mathbf{Z}_i\) = Observed characteristics.
- \(\mathbf{\mu}_i\) = Unobserved factors.
- \(\epsilon_{it}\) = Transitory shocks (random noise).
To ensure a valid synthetic control, the weights \(\mathbf{W}^*\) must satisfy:
\[ \sum_{j=2}^{J+1} w_j^* \mathbf{Z}_j = \mathbf{Z}_1 \]
\[ \sum_{j=2}^{J+1} w_j^* Y_{j1} = Y_{11}, \quad \dots, \quad \sum_{j=2}^{J+1} w_j^* Y_{jT_0} = Y_{1T_0} \]
This guarantees that the synthetic control closely matches the treated unit in pre-treatment periods.
Bias Considerations:
- The accuracy of SCM depends on the ratio of transitory shocks (\(\epsilon_{it}\)) to pre-treatment periods (\(T_0\)). In other words, you should have good fit for \(Y_{1t}\) for pre-treatment period (i.e., \(T_0\) should be large while small variance in \(\epsilon_{it}\))
- Good fit in pre-treatment periods (large \(T_0\)) is crucial.
- If the pre-treatment fit is poor, bias correction methods are required Ben-Michael, Feller, and Rothstein (2020).
References
Abadie, Alberto, Alexis Diamond, and Jens Hainmueller. 2010. “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.
Ben-Michael, Eli, Avi Feller, and Jesse Rothstein. 2020. “Varying Impacts of Letters of Recommendation on College Admissions: Approximate Balancing Weights for Subgroup Effects in Observational Studies.” arXiv Preprint arXiv:2008.04394.