32.13 Using Multiple Outcomes to Improve the Synthetic Control Method

Typically, SCM constructs a weighted combination of untreated control units to approximate the counterfactual outcome of the treated unit. However, standard SCM is limited to a single outcome variable, which can lead to biased estimates when multiple correlated outcomes are available.

In their work, L. Sun, Ben-Michael, and Feller (2023) propose a novel extension of SCM that leverages multiple outcome variables to improve causal inference by:

1. Using a common set of synthetic control weights across all outcomes rather than estimating separate weights for each outcome.
2. Reducing bias using a low-rank factor model, which exploits shared latent structures across outcomes.

32.13.1 Standard Synthetic Control Method

Let $Y_{itk}$ denote the observed outcome for unit $i$ at time $t$ for outcome $k$, where $i = 1, \dots, N$, $t = 1, \dots, T$, and $k = 1, \dots, K$. The potential outcomes framework assumes:

$$Y_{itk}(d) = \mu_{itk} + \delta_{itk} d + \varepsilon_{itk}, \quad d \in \{0,1\}$$

where:

• $\mu_{itk}$ represents the latent structure of untreated outcomes.

• $\delta_{itk}$ is the treatment effect.

• $\varepsilon_{itk} \sim \mathcal{N}(0, \sigma^2)$ is random noise.

For unit $i=1$ (the treated unit), the observed outcome follows:

$$Y_{1tk} = Y_{1tk}(0) + D_{1t} \delta_{1tk}$$

where $D_{1t}$ is an indicator for treatment at time $t$. The challenge is to estimate the counterfactual outcome $Y_{1tk}(0)$, which is unobserved post-treatment.

SCM estimates $Y_{1tk}(0)$ as a weighted combination of control units:

$$\hat{Y}_{1tk}(0) = \sum_{i=2}^{N} w_i Y_{itk}$$

where weights $w_i$ are chosen to minimize pre-treatment imbalance.

32.13.2 Using Multiple Outcomes for Bias Reduction

Instead of estimating separate weights $w_k$ for each outcome $k$, L. Sun, Ben-Michael, and Feller (2023) propose a single set of weights $w$ across all outcomes. This approach is justified under a low-rank factor model, which assumes that multiple outcomes share common latent factors.

32.13.2.1 Low-Rank Factor Model

Assume the untreated potential outcome follows a linear factor structure:

$$Y_{itk}(0) = X_{it} \beta_k + \lambda_i' f_{tk} + \varepsilon_{itk}$$

where:

• $X_{it}$ are observed covariates.

• $\beta_k$ are outcome-specific coefficients.

• $\lambda_i$ are unit-specific factor loadings.

• $f_{tk}$ are time-and-outcome-specific latent factors.

If all outcomes share the same latent factor structure, then the bias in synthetic control estimation can be reduced by a factor of $1 / \sqrt{K}$ as the number of outcomes $K$ increases.

32.13.3 Estimation Methods

L. Sun, Ben-Michael, and Feller (2023) propose two methods for constructing a common synthetic control:

1. Concatenated Outcome Weights: Estimate weights by minimizing imbalance across all outcomes simultaneously:

   $$\hat{w} = \arg\min_w \sum_{k=1}^{K} || Y_{1,\text{pre},k} - \sum_{i=2}^{N} w_i Y_{i,\text{pre},k} ||^2$$

2. Averaged Outcome Weights: Estimate weights based on a linear combination (e.g., average) of outcomes:

   $$\hat{w} = \arg\min_w || \frac{1}{K} \sum_{k=1}^{K} Y_{1,\text{pre},k} - \sum_{i=2}^{N} w_i \frac{1}{K} \sum_{k=1}^{K} Y_{i,\text{pre},k} ||^2$$

These methods improve SCM performance by reducing variance and overfitting to noise.

32.13.4 Empirical Application: Flint Water Crisis

To illustrate the benefits of multiple outcome SCM, L. Sun, Ben-Michael, and Feller (2023) re-analyze the Flint water crisis, which led to lead contamination in drinking water, potentially affecting student performance.

Four key educational outcomes were studied:

1. Math Achievement
2. Reading Achievement
3. Special Needs Status
4. Daily Attendance

By applying common weights across these outcomes, their SCM results showed:

• Reduced bias and improved robustness compared to separate SCM fits.

• Better pre-treatment fit for educational outcomes.

• Stronger evidence of educational impacts following the crisis.

# Load necessary libraries
library(augsynth)

# Fit SCM using a common set of weights across multiple outcomes
synth_model <- augsynth_multiout()

References

Sun, Liyang, Eli Ben-Michael, and Avi Feller. 2023. “Using Multiple Outcomes to Improve the Synthetic Control Method.” arXiv Preprint arXiv:2311.16260.