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.

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.


References

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