32.10 Synthetic Control with Staggered Adoption

While traditional SCM focuses on cases with a single treated unit, many real-world policies exhibit staggered adoption, where different units receive treatment at different times.

In staggered adoption designs, policy implementation occurs over multiple time periods across different units (e.g., states, companies, or regions). This presents challenges:

• Traditional SCM limitations: SCM was designed for a single treated unit and does not naturally accommodate multiple adoption times.
• Heterogeneous treatment effects: The impact of the intervention may vary over time or across units.
• Estimation bias: Common approaches such as Two-Way Fixed Effects can yield biased results when treatment effects are heterogeneous.

32.10.1 Partially Pooled Synthetic Control

Ben-Michael, Feller, and Rothstein (2022) propose a partially pooled SCM approach, balancing trade-offs between separate SCM for each unit and a fully pooled approach that estimates a single synthetic control for all treated units. The key ideas include:

1. Two imbalance measures:
   • Individual unit-level imbalance.
   • Aggregate imbalance for the average treated unit.
2. Optimization framework:
   • Weights are chosen to minimize a weighted sum of these two imbalance measures.
   • This method reduces bias compared to estimating separate SCM models for each unit​.

Mathematically, let:

• $Y_{it}$ be the outcome of interest for unit $i$ at time $t$.

• $T_i$ be the treatment adoption time for unit $i$.

• $W_i$ be the synthetic control weight assigned to unit $i$.

Then, for treated unit $j$, the estimated counterfactual is:

$$\hat{Y}_{jT_j+k} = \sum_{i=1}^{N} \hat{W}_{ij} Y_{iT_j+k}$$

The objective function combines both individual and aggregate balance constraints:

$$\min_W \lambda \sum_{j} ||Y_{j,\text{pre}} - \sum_i W_{ij} Y_{i,\text{pre}}||^2 + (1-\lambda) ||Y_{\text{treated, pre}} - \sum_i W_i Y_{i,\text{pre}}||^2$$

where $\lambda$ is a tuning parameter that controls the trade-off between individual and pooled balance​.

# Load necessary library
library(augsynth)

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References

———. 2022. “Synthetic Controls with Staggered Adoption.” Journal of the Royal Statistical Society Series B: Statistical Methodology 84 (2): 351–81.