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The synthetic control (cohort) method is a very promising approach to causal inference that has been used in a number of interesting studies. It's particularly useful in situations where data are only available in aggregate or there is only one treatment observed.

In Abadie et al (2015), the authors claim that unobserved, time-varying confounders are essentially control for given an argument based on intuition:

"The intuition of this result is straightforward: Only units that are alike in both observed and unobserved determinants of the outcome variable as well as in the effect of those determinants on the outcome variable should produce similar trajectories of the outcome variable over extended periods of time."

I buy this argument but I'd like to see a firmer exposition of why this is the case. If one selects on pre-treatment trends in outcomes and observed factors relevant in driving outcomes, how does the synthetic control method control for both unobserved AND time-varying variables that may also influence outcomes?

Andy B
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Great question. The technicalities are probably buried in the appendices of Abadie et al. 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 (2010): 493-505.

However, there is a recent and simpler to read paper by Xu [1] that generalizes the synthetic control method. It has two central assumptions:

Assumption 1: $$Y_{it} = \delta_{it}D_{it} + x'_{it}\beta + \lambda'_if_t + \epsilon_{it} $$ and (part of) assumption 2:

$$ \epsilon_{it} \perp D_{js}|x_{js}, \lambda_j, f_s $$

The unobserved confounders are $\lambda'_if_t$ and $\epsilon_{it}$. We need to assume that the treatment is independent from $\epsilon_{it}$, which are unobserved confounders that vary across time and across units. This is similar to standard panel models.

However, the argument goes that with the synthetic control method we do not need to assume that treatment is independent from $\lambda'_if_t$, and that the latter can capture "a wide range" of variables that with standard panel models would be contained in $\epsilon_{it}$. Xu 2017 (p. 62) gives the following example (I've added the corresponding symbols):

Suppose a law is passed in a state because the public opinion in that state becomes more liberal. Because changing ideologies are often cross-sectionally correlated across states, a latent factor [$f_t$] may be able to capture shifting ideology at the national level; the national shifts may have a larger impact on a state that has a tradition of mass liberalism or has a higher proportion of manufacturing workers than a state that is historically conservative [$\lambda'_i$]. Controlling for this unobserved confounder, therefore, can alleviate the concern that the passage of the law is endogenous to changing ideology of a state’s constituents to a great extent.

[1] Xu, Y. (2017). Generalized Synthetic Control Method: Causal Inference with Interactive Fixed Effects Models. Political Analysis, 25(1), 57-76.

Julian Schuessler
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