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Synthetic Control

The model extends the traditional linear panel data (difference-in-differences) framework, allowing that the effects of unobserved variables on the outcome vary with time. (Abadie & Diamond & Hainmueller (2010))

-> This is the key differnce to the difference-in-difference design. However, it is important to clarify that this statement refers to !time-constant! unobserved confounders. Now, the intuition that reproducing well a long time-series of pre-treatment outcomes of the eventually treated unit with a weighted average of the donor pool also picks up the effect of unobserved confounders. Then, because these are time-constant, their time-varying effect after treatment is also incporporated.

Consider the following factor model:

\begin{align*} Y^N_{it} = \delta_t + \mathbb{\theta}_t Z_{i} + \lambda_t \mu_i + \epsilon_{it} \end{align*}

If \(\lambda_t = \lambda\), i.e. \(\lambda_t\) is constant over time, then we are back in the standard setting.

References

• Abadie, A. (2021). Using synthetic controls: feasibility, data requirements, and methodological aspects, Journal of Economic Literature, 59(2), 391-425.

• Abadie, A., Diamond, A., & Hainmueller, J. (2010). Synthetic control methods for comparative casecStudies: cstimating the effect of california’s tobacco control program, Journal of the American Statistical Association, 105(490), 493-505.

• Firpo, S., & Possebom, V. (2018). Synthetic control method: inference, sensitivity analysis and confidence sets, Journal of Causal Inference, 6(2), 2-26.