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.

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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 $t$.

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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.