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Functional Synthetic Control Methods for Metric Space-Valued Outcomes

Ryo Okano, Daisuke Kurisu

TL;DR

This paper extends causal inference with synthetic controls to outcomes that lie in general metric spaces by embedding them isometrically into Hilbert spaces. The authors introduce the Functional Synthetic Control (FSC) method and a bias-corrected, ridge-augmented FSC, derive finite-sample error bounds under autoregressive and latent-factor models, and develop conformal inference and placebo tests for counterfactuals. They demonstrate improved pre-treatment fit and reliable inference through simulations and three empirical analyses: abortion legislation's impact on fertility curves in East Germany, the Soviet collapse’s effect on Russia’s mortality distributions, and the Brexit announcement’s effect on UK service-trade covariance. The framework accommodates diverse data types (functions, distributions, covariance matrices, networks, compositional data) and provides practical tools for causal analysis in complex-valued outcomes with scalable theory. The work opens avenues for extensions to SDID and staggered adoption designs and includes publicly available code.

Abstract

The synthetic control method (SCM) is a widely used tool for evaluating causal effects of policy changes in panel data settings. Recent studies have extended its framework to accommodate complex outcomes that take values in metric spaces, such as distributions, functions, networks, covariance matrices, and compositional data. However, due to the lack of linear structure in general metric spaces, theoretical guarantees for estimation and inference within these extended frameworks remain underdeveloped. In this study, we propose the functional synthetic control (FSC) method as an extension of the SCM for metric space-valued outcomes. To address challenges arising from the nonlinearlity of metric spaces, we leverage isometric embeddings into Hilbert spaces. Building on this approach, we develop the FSC and augmented FSC estimators for counterfactual outcomes, with the latter being a bias-corrected version of the former. We then derive their finite-sample error bounds to establish theoretical guarantees for estimation, and construct prediction sets based on these estimators to conduct inference on causal effects. We demonstrate the usefulness of the proposed framework through simulation studies and three empirical applications.

Functional Synthetic Control Methods for Metric Space-Valued Outcomes

TL;DR

This paper extends causal inference with synthetic controls to outcomes that lie in general metric spaces by embedding them isometrically into Hilbert spaces. The authors introduce the Functional Synthetic Control (FSC) method and a bias-corrected, ridge-augmented FSC, derive finite-sample error bounds under autoregressive and latent-factor models, and develop conformal inference and placebo tests for counterfactuals. They demonstrate improved pre-treatment fit and reliable inference through simulations and three empirical analyses: abortion legislation's impact on fertility curves in East Germany, the Soviet collapse’s effect on Russia’s mortality distributions, and the Brexit announcement’s effect on UK service-trade covariance. The framework accommodates diverse data types (functions, distributions, covariance matrices, networks, compositional data) and provides practical tools for causal analysis in complex-valued outcomes with scalable theory. The work opens avenues for extensions to SDID and staggered adoption designs and includes publicly available code.

Abstract

The synthetic control method (SCM) is a widely used tool for evaluating causal effects of policy changes in panel data settings. Recent studies have extended its framework to accommodate complex outcomes that take values in metric spaces, such as distributions, functions, networks, covariance matrices, and compositional data. However, due to the lack of linear structure in general metric spaces, theoretical guarantees for estimation and inference within these extended frameworks remain underdeveloped. In this study, we propose the functional synthetic control (FSC) method as an extension of the SCM for metric space-valued outcomes. To address challenges arising from the nonlinearlity of metric spaces, we leverage isometric embeddings into Hilbert spaces. Building on this approach, we develop the FSC and augmented FSC estimators for counterfactual outcomes, with the latter being a bias-corrected version of the former. We then derive their finite-sample error bounds to establish theoretical guarantees for estimation, and construct prediction sets based on these estimators to conduct inference on causal effects. We demonstrate the usefulness of the proposed framework through simulation studies and three empirical applications.
Paper Structure (37 sections, 16 theorems, 176 equations, 23 figures, 3 tables)

This paper contains 37 sections, 16 theorems, 176 equations, 23 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Setup
  3. Outcomes in Metric Space and Causal Effects
  4. Examples
  5. Functional Synthetic Control Method and Its Augmentation
  6. Functional Synthetic Control Method
  7. Augmented Functional Synthetic Control Method
  8. General form.
  9. Ridge augmented FSC method.
  10. Finite-Sample Error Bounds and Inference
  11. Error Bounds under Autoregressive Model
  12. Error Bounds under Latent Factor Model
  13. Inference
  14. Conformal inference for counterfactual outcomes.
  15. Placebo permutation test.
  16. ...and 22 more sections

Key Result

Lemma 1

For any $t=T_0+1, \ldots, T$, the ridge augmented FSC estimator eq:est_ridge_y is expressed as where the weights $\hat{\gamma}^{\mathrm{aug}} = (\hat{\gamma}_i^{\mathrm{aug}})_{i=2}^N$ are given by Moreover, the weights $\hat{\gamma}^{\mathrm{aug}}$ are the solution to the following constrained optimization problem:

Figures (23)

  • Figure 1: Boxplots of estimation errors under the autoregressive model with low noise (left), medium noise (middle), and high noise (right). AFSC (i), AFSC (ii), and AFSC (iii) denote the ridge augmented FSC with penalty hyperparameters $\lambda = 100\hat{\lambda}_{\text{cv}},\ \hat{\lambda}_{\text{cv}},\ \text{and}\ 0.01\hat{\lambda}_{\text{cv}}$, where $\hat{\lambda}_{\text{cv}}$ is the value of $\lambda$ selected by cross validation.
  • Figure 2: Boxplots of estimation errors under the latent factor model with low noise (left), medium noise (middle), and high noise (right).
  • Figure 3: Observed ASFR curves for East Germany and the corresponding ASFR curves for the FSC and ridge-augmented FSC units during pre-treatment periods (1964–1971) and the post-treatment periods (1972–1975).
  • Figure 4: Estimates and pointwise 90% prediction bands for the causal effects ${Y}_{1t}^I - {Y}_{1t}^N, t = 1972, 1973, 1974, 1975$ based on the ridge augmented FSC.
  • Figure 5: The quantile functions of the observed age-at-death distributions for Russia and the corresponding quantile functions obtained from the FSC and ridge augmented FSC units. The priods from 1985 to 1990 are pre-treatment priods, while 1991 to 1999 are the post-treatment periods.
  • ...and 18 more figures

Theorems & Definitions (28)

  • Example 1: Functional data
  • Example 2: One-dimensional probability distributions
  • Example 3: Symmetric positive semidefinite matrices
  • Example 4: Networks
  • Example 5: Compositional data
  • Remark 1: Isometric embedding of metric spaces into Hilbert spaces
  • Remark 2: Connection with the geodesic synthetic control estimator
  • Remark 3: Connection with the augmented geodesic synthetic control estimator
  • Remark 4: Modification via rearrangement method
  • Lemma 1
  • ...and 18 more