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 \(t\).