32.9 Augmented Synthetic Control Method

The Augmented Synthetic Control Method (ASCM), introduced by Ben-Michael, Feller, and Rothstein (2021), extends the Synthetic Control Method to cases where perfect pre-treatment fit is infeasible. ASCM combines SCM weighting with bias correction through an outcome model, improving estimates when SCM alone fails to match pre-treatment outcomes precisely.

Key Idea:

• Standard SCM requires that the synthetic control closely matches the treated unit in pre-treatment periods.
• When this is not possible, ASCM adjusts for bias using outcome modeling, similar to bias correction in matching estimators.
• ASCM can be seen as a trade-off between SCM and regression-based approaches, incorporating both synthetic control weighting and outcome modeling.

ASCM builds on SCM but relaxes its strong convex hull assumption. Key assumptions:

• No Interference: Treatment affects only the treated unit.

• No Unobserved Time-Varying Confounders: Changes over time should not be correlated with treatment assignment.

• Regularization Controls Extrapolation Bias: Ridge penalty prevents overfitting.

ASCM is recommended when:

1. SCM alone does not provide a good pre-treatment fit.

2. Only one treated unit is available.

3. Auxiliary covariates need to be incorporated.

Advantages of ASCM

1. Handles Poor Pre-Treatment Fit
   • Standard SCM fails when the treated unit lies outside the convex hull of donor units.
   • ASCM allows negative weights (via ridge regression) to improve fit.
2. Balances Bias and Variance
   • Ridge penalty controls extrapolation, reducing overfitting.
3. Flexible Estimation Framework
   • Works with auxiliary covariates, extending beyond pure pre-treatment matching.

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Let:

• $J + 1$ units be observed over $T$ time periods.

• The first unit ($i=1$) is treated in periods $T_0 + 1, \dots, T$. - The remaining $J$ units are the donor pool (potential controls).

• Define:

  • $Y_{it}^I$: Outcome for unit $i$ under treatment.

  • $Y_{it}^N$: Outcome for unit $i$ in the absence of treatment (counterfactual).

The treatment effect of interest:

$$\tau_{1t} = Y_{1t}^I - Y_{1t}^N$$

where:

$$Y_{1t}^I = Y_{1t}$$

but $Y_{1t}^N$ is unobserved and must be estimated.

ASCM improves SCM by incorporating an outcome model to correct for poor pre-treatment fit. The counterfactual outcome is estimated as:

$$\hat{Y}^{\text{aug}}_{1T}(0) = \sum_{i=2}^{J+1} w_i Y_{iT} + \left( m_1 - \sum_{i=2}^{J+1} w_i m_i \right)$$

where:

• $w_i$ are SCM weights chosen to best match pre-treatment outcomes.

• $m_i$ is an outcome model prediction for unit $i$.

• If SCM achieves perfect pre-treatment fit, $m_1 - \sum w_i m_i \approx 0$, and ASCM reduces to standard SCM.

The most common implementation, Ridge ASCM, uses ridge regression to estimate $m_i$, leading to:

$$\hat{Y}^{\text{aug}}_{1T}(0) = \sum_{i=2}^{J+1} w_i Y_{iT} + \left( X_1 - \sum w_i X_i \right) \beta$$

where $\beta$ is estimated using ridge regression of post-treatment outcomes on pre-treatment outcomes.

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References

———. 2021. “The Augmented Synthetic Control Method.” Journal of the American Statistical Association 116 (536): 1789–1803.