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:
SCM alone does not provide a good pre-treatment fit.
Only one treated unit is available.
Auxiliary covariates need to be incorporated.
Advantages of ASCM
- 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.
- Balances Bias and Variance
- Ridge penalty controls extrapolation, reducing overfitting.
- Flexible Estimation Framework
- Works with auxiliary covariates, extending beyond pure pre-treatment matching.
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.