32.6 Estimation

We observe \(J + 1\) units over \(T\) time periods.

  • The first unit (\(i = 1\)) is treated starting from time \(T_0 + 1\).
  • The remaining \(J\) units serve as the donor pool (potential controls).
  • Define:
    • \(Y_{it}^I\): Outcome for unit \(i\) under treatment (\(i=1\), for \(t \geq T_0 + 1\)).
    • \(Y_{it}^N\): Outcome for unit \(i\) in the absence of treatment (counterfactual).

The goal is to estimate the treatment effect:

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

where we observe:

\[ Y_{1t}^I = Y_{1t} \]

but \(Y_{1t}^N\) is unobserved and must be estimated using a synthetic control.

32.6.1 Constructing the Synthetic Control

To estimate the counterfactual outcome, we create a synthetic control unit, a weighted combination of the untreated donor units. We assign weights \(\mathbf{W} = (w_2, \dots, w_{J+1})'\) that satisfy:

  • Non-negativity constraint:
    \[ w_j \geq 0, \quad \forall j = 2, \dots, J+1 \]
  • Sum-to-one constraint:
    \[ w_2 + w_3 + \dots + w_{J+1} = 1 \]

The optimal weights are found by solving:

\[ \min_{\mathbf{W}} ||\mathbf{X}_1 - \mathbf{X}_0 \mathbf{W}|| \]

where:

  • \(\mathbf{X}_1\) is a \(k \times 1\) vector of pre-treatment characteristics for the treated unit.
  • \(\mathbf{X}_0\) is a \(k \times J\) matrix of pre-treatment characteristics for the donor units.

A common approach is to minimize the weighted sum:

\[ \min_{\mathbf{W}} \sum_{h=1}^{k} v_h (X_{h1} - w_2 X_{h2} - \dots - w_{J+1} X_{hJ+1})^2 \]

where:

  • \(v_h\) represents the predictive power of each \(k\)-dimensional pre-treatment characteristic on \(Y_{1t}^N\).
  • The weights \(v_h\) can be chosen either:
    • Explicitly by the researcher, or
    • Data-driven via optimization.

32.6.2 Penalized Synthetic Control

To reduce interpolation bias, the penalized synthetic control method (Abadie and L’hour 2021) modifies the optimization problem:

\[ \min_{\mathbf{W}} ||\mathbf{X}_1 - \sum_{j=2}^{J+1}W_j \mathbf{X}_j ||^2 + \lambda \sum_{j=2}^{J+1} W_j ||\mathbf{X}_1 - \mathbf{X}_j||^2 \]

where:

  • \(\lambda > 0\) controls the trade-off between fit and regularization:
    • \(\lambda \to 0\): Standard synthetic control (unpenalized).
    • \(\lambda \to \infty\): Nearest-neighbor matching (strong penalization).
  • This method ensures:
    • Sparse and unique solutions for weights.
    • Exclusion of dissimilar control units (reducing interpolation bias).

The final synthetic control estimator is:

\[ \hat{\tau}_{1t} = Y_{1t} - \sum_{j=2}^{J+1} w_j^* Y_{jt} \]

where \(Y_{jt}\) is the outcome for unit \(j\) at time \(t\).


References

Abadie, Alberto, and Jérémy L’hour. 2021. “A Penalized Synthetic Control Estimator for Disaggregated Data.” Journal of the American Statistical Association 116 (536): 1817–34.