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).

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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.