Synthetic Control Extensions

Summary

Section 8 of Abadie (2021) surveys recent extensions to the canonical synthetic control estimator. The most practically important is the penalized synthetic control (Abadie and L’Hour 2019) for multiple treated units, which ensures unique and sparse weights while controlling interpolation biases. Other extensions include bias-corrected estimators, regression-based methods that allow extrapolation (Doudchenko and Imbens 2016), and matrix completion methods (Amjad et al. 2018; Athey et al. 2020) that handle missing data and high-dimensional settings.

Overview

The canonical synthetic control (one treated unit, convex combination, permutation inference) is well-suited for comparative case studies with a single treated aggregate unit. Recent years have seen substantial methodological development extending the framework to: (i) multiple simultaneously treated units, (ii) settings where the convex hull condition fails, (iii) regression-based generalizations, and (iv) matrix completion methods for high-dimensional panel data.

Multiple Treated Units: Penalized Synthetic Control

When $I>1$ units are treated simultaneously, fitting a separate synthetic control for each treated unit is straightforward but creates new challenges: the minimizer of the weight optimization may not be unique (especially when $k$ is moderate), and each treated unit may be matched to donors that are far away in the predictor space.

Definition: Penalized Synthetic Control Estimator (Abadie and L'Hour 2019, Eq. 13)

With $I$ treated units ($j=1,\dots ,I$) and $J$ untreated units ($j=I+1,\dots ,I+J$), the penalized synthetic control minimizes for each treated unit $i$:

$$\Vert {\mathbf{X}}_i-{\mathbf{X}}_0\mathbf{W}{\Vert }^2+\lambda \sum _{j=I+1}^{I+J}{w}_j\Vert {\mathbf{X}}_i-{\mathbf{X}}_j{\Vert }^2$$

subject to weights nonneg. and summing to 1.

The first term is the aggregate discrepancy between the treated unit and its synthetic control. The second term penalizes the pairwise matching discrepancy — the distance between the treated unit and each donor weighted by that donor’s contribution.

As $\lambda \to \infty$: converges to one-to-one matching (nearest neighbor) As $\lambda \to 0$: reduces to the unpenalized synthetic control

Theorem: Uniqueness and Sparsity (Abadie and L'Hour 2019)

If $\lambda >0$ and the columns of ${\mathbf{X}}_0$ are in general quadratic position, the minimizer of the penalized objective is unique and sparse — resolving the non-uniqueness problem that arises with multiple treated units in the canonical formulation.

The treatment effect for each treated unit $i$ at $t>T_0$ is:

$${\hat{\tau }}_{it}=Y_{it}-\sum _{j=I+1}^{I+J}{w}_{ij}^{\ast }{Y}_{jt}$$

And the average treatment effect across treated units:

$${\hat{\tau }}_t=\frac{1}{I}\sum _{i=1}^I{\hat{\tau }}_{it}$$

Cross-validation techniques in Abadie and L’Hour (2019) select the penalty parameter $\lambda$.

Bias-Corrected Synthetic Control

When the synthetic control cannot perfectly reproduce the treated unit’s predictor values (large ${\mathbf{X}}_1-{\mathbf{X}}_0{\mathbf{W}}^{\ast }$), regression adjustments can attenuate the resulting bias.

Definition: Bias-Corrected Synthetic Control (Ben-Michael et al. 2020, Eq. 15–16)

Let ${\hat{\mu }}_{0t}(\cdot )$ be a (parametric or nonparametric) regression of untreated outcomes $Y_{I+1,t},\dots ,Y_{I+J,t}$ on predictor values for the untreated units. The bias-corrected estimator is:

$${\hat{\tau }}_{it}=\left(Y_{it}-{\hat{\mu }}_{0t}({\mathbf{X}}_i)\right)-\sum _{j=I+1}^{I+J}{w}_{ij}^{\ast }\left(Y_{jt}-{\hat{\mu }}_{0t}({\mathbf{X}}_j)\right)$$

Equivalently: the standard synthetic control estimator applied to regression residuals rather than raw outcomes.

This is closely related to the augmented synthetic control (Ben-Michael, Feller, and Rothstein 2020) and connects to the doubly-robust approach in Local Average Treatment Effects and causal inference more broadly. The bias correction is most valuable when the treated unit falls outside the convex hull of the donor pool.

Regression-Based Methods and Extrapolation

Doudchenko and Imbens (2016) generalize synthetic controls by relaxing the convex combination constraint, allowing extrapolation:

Definition: Elastic Net Synthetic Control (Doudchenko and Imbens 2016, Eq. 17–18)

Minimize over $(\alpha ,w_2,\dots ,w_{J+1})\in {\mathbb{R}}^{J+1}$:

$$\sum _{t=1}^{T_0}{\left(Y_{1t}-\alpha -\sum _{j=2}^{J+1}{w}_j{Y}_{jt}\right)}^2+{\lambda }_1\left(\frac{1-{\lambda }_2}{2}\sum _{j=2}^{J+1}{w}_j^2+{\lambda }_2\sum _{j=2}^{J+1}|w_j|\right)$$

where ${\lambda }_1\ge 0$, $0\le {\lambda }_2\le 1$ are regularization parameters.

The intercept $\alpha$ allows a constant shift between ${\mathbf{X}}_1$ and ${\mathbf{X}}_0$, handling cases where the treated unit’s level of the outcome differs from any convex combination of donor units.

Parameters selected by cross-validation.

The counterfactual estimate is ${\hat{Y}}_{1t}^N=\hat{\alpha }+{\sum }_{j=2}^{J+1}{\hat{w}}_j{Y}_{jt}$.

Trade-off with canonical SC:

• Elastic net allows extrapolation → useful when convex hull condition fails
• But: allowing negative weights makes the counterfactual less interpretable and may hide interpolation biases
• Canonical SC’s non-negativity constraint makes the counterfactual transparent and limits extrapolation

Matrix Completion Methods

For panel data with many units and time periods (including settings with missing outcomes), matrix completion methods offer a flexible alternative.

Definition: Matrix Completion for Synthetic Control (Amjad et al. 2018; Athey et al. 2020)

Model the matrix of untreated potential outcomes $\{Y_{jt}^N\}$ as generated by a nonlinear factor structure:

$$Y_{jt}^N=f({\mathbf{\mu }}_j,{\mathbf{\lambda }}_t)+{\epsilon }_{jt}$$

where $f$ is unknown. Assume the matrix $\{M_{jt}\}=\{f({\mathbf{\mu }}_j,{\mathbf{\lambda }}_t)\}$ is low-rank.

Estimate $\{M_{jt}\}$ via matrix completion (e.g., singular value thresholding). Use $\{{\hat{M}}_{jt}\}$ to:

1. Impute missing potential outcomes
2. Estimate untreated potential outcomes for treated units in post-intervention periods
3. Compute synthetic controls as linear combinations of ${\hat{M}}_{jt}$ values

Extensions:

• Amjad et al. (2019): incorporates additional covariates ${\mathbf{Z}}_j$ alongside outcomes
• Athey et al. (2020): postulates $Y_{jt}^N=M_{jt}+{\epsilon }_{jt}$ with low-rank $\{M_{jt}\}$; handles missing entries via matrix estimation; naturally accommodates multiple treated units in post-intervention periods

Advantage over canonical SC: Handles settings with many units and time periods, including high-dimensional micro-data panels. Does not require the treated unit to be an aggregate entity.

Limitation: The low-rank assumption is less transparent than the linear factor model; model validity is harder to assess.

Inference Extensions

Hahn and Shi (2017): Propose applying the end-of-sample instability test (Andrews 2003) as an inferential procedure for synthetic controls with stationary data and large $T_0$. Related to the backdating diagnostic.

Chernozhukov, Wüthrich, and Zhu (2019a, 2019b): Devise a sampling-based inferential procedure using permutations of regression residuals in the time dimension. Provides valid inference without the linear factor model assumption. Propose bias-corrected synthetic control with confidence intervals centered on the K-fold cross-fitted estimate.

Cattaneo, Feng, and Titiunik (2021): Predictive intervals for ${\hat{\tau }}_{it}$ that take into account both estimation uncertainty in the untreated potential outcome model and irreducible uncertainty from the unobserved random error $u_t$.

Connections

• Synthetic Control Bias Theory: The penalized estimator addresses the non-uniqueness that arises in the canonical estimator when $k$ is large
• Differences-in-Differences: Elastic net and matrix completion methods blur the boundary between DiD (regression on panel data) and synthetic control (matching on pre-treatment trajectory)
• Nonparametric Causal Inference: Matrix completion via BART offers a related approach to panel counterfactual estimation from the Bayesian nonparametric perspective
• Bayesian Difference in Differences: PyMC implementation of Bayesian counterfactual inference for aggregate time-series; complementary to synthetic control for the same setting

Generalized Synthetic Control (Xu 2017)

The Generalized Synthetic Control Method (GSC) is the primary practical recommendation for multiple treated units. It directly estimates the linear factor model on control group data, then projects treated units’ pre-treatment outcomes onto the estimated factor space. Unlike the penalized SC above, GSC:

• Produces frequentist SEs and CIs via parametric bootstrap (not permutation inference)
• Selects the number of factors via built-in cross-validation
• Is implemented in the gsynth R package

See Generalized Synthetic Control Method for the full treatment.

See Also

• Synthetic Control — the canonical estimator
• Generalized Synthetic Control Method — the primary extension for multiple treated units (Xu 2017)
• Synthetic Control Bias Theory — the formal theory motivating the penalized estimator
• Synthetic Control Inference and Diagnostics — permutation inference, generalized to multiple units
• Differences-in-Differences — panel regression methods that matrix completion bridges to
• Abadie 2021 - Overview — full paper overview
• The Selection Problem — the fundamental challenge that synthetic control (and these extensions) addresses by constructing a credible counterfactual