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Synthetic Regressing Control

Rong J. B. Zhu

TL;DR

This paper introduces Synthetic Regressing Control (SRC), a method to improve causal effect estimation in panel data by first aligning pre-treatment trajectories through unit-wise regressions and then synthesizing regressed controls with weights chosen by an unbiased risk estimator. SRC directly addresses two SC limitations: interpolation bias from imperfect pre-treatment fit and the noise-related extrapolation risk associated with the unit-sum constraint, while preserving the interpretability of weighted donor contributions. The authors prove asymptotic optimality of SRC for out-of-sample predictions and demonstrate superior or competitive performance relative to standard SC variants and OLS in simulations and a Basque country GDP application. Extensions include high-dimensional screening of donor units and incorporating auxiliary covariates, broadening SRC’s applicability to settings with many controls and covariates, with practical guidance for implementation and inference. Overall, SRC offers a principled, data-driven way to balance pre-treatment fit and extrapolation risk in synthetic control analyses, with demonstrated gains in predictive accuracy.

Abstract

Estimating weights in the synthetic control method, typically resulting in sparse weights where only a few control units have non-zero weights, involves an optimization procedure that selects and combines control units to closely match the treated unit. However, it is not uncommon for the linear combination of pre-treatment period outcomes for the control units, using nonnegative weights with the constraint that their sum equals one, to inadequately approximate the pre-treatment outcomes for the treated unit. To address the issue, this paper proposes a simple and effective method called Synthetic Regressing Control (SRC). The SRC method begins by performing the univariate linear regression to appropriately align the pre-treatment periods of the control units with the treated unit. Subsequently, a SRC estimator is obtained by synthesizing the regressed controls. To determine the weights in the synthesis procedure, we propose an approach that utilizes a criterion of an unbiased risk estimator. Theoretically, we show that the synthesis way is asymptotically optimal in the sense of achieving the minimum loss of the infeasible best possible synthetic estimator. Extensive numerical experiments highlight the advantages of the SRC method.

Synthetic Regressing Control

TL;DR

This paper introduces Synthetic Regressing Control (SRC), a method to improve causal effect estimation in panel data by first aligning pre-treatment trajectories through unit-wise regressions and then synthesizing regressed controls with weights chosen by an unbiased risk estimator. SRC directly addresses two SC limitations: interpolation bias from imperfect pre-treatment fit and the noise-related extrapolation risk associated with the unit-sum constraint, while preserving the interpretability of weighted donor contributions. The authors prove asymptotic optimality of SRC for out-of-sample predictions and demonstrate superior or competitive performance relative to standard SC variants and OLS in simulations and a Basque country GDP application. Extensions include high-dimensional screening of donor units and incorporating auxiliary covariates, broadening SRC’s applicability to settings with many controls and covariates, with practical guidance for implementation and inference. Overall, SRC offers a principled, data-driven way to balance pre-treatment fit and extrapolation risk in synthetic control analyses, with demonstrated gains in predictive accuracy.

Abstract

Estimating weights in the synthetic control method, typically resulting in sparse weights where only a few control units have non-zero weights, involves an optimization procedure that selects and combines control units to closely match the treated unit. However, it is not uncommon for the linear combination of pre-treatment period outcomes for the control units, using nonnegative weights with the constraint that their sum equals one, to inadequately approximate the pre-treatment outcomes for the treated unit. To address the issue, this paper proposes a simple and effective method called Synthetic Regressing Control (SRC). The SRC method begins by performing the univariate linear regression to appropriately align the pre-treatment periods of the control units with the treated unit. Subsequently, a SRC estimator is obtained by synthesizing the regressed controls. To determine the weights in the synthesis procedure, we propose an approach that utilizes a criterion of an unbiased risk estimator. Theoretically, we show that the synthesis way is asymptotically optimal in the sense of achieving the minimum loss of the infeasible best possible synthetic estimator. Extensive numerical experiments highlight the advantages of the SRC method.
Paper Structure (19 sections, 2 theorems, 84 equations, 1 figure, 8 tables, 4 algorithms)

This paper contains 19 sections, 2 theorems, 84 equations, 1 figure, 8 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Synthetic Control Method
  4. Overview
  5. Oversights in the SC Method
  6. Synthetic Regressing Control
  7. A Model Framework
  8. Rationale for the Unit Regression Step
  9. Synthetic Regressing Control
  10. Determination of optimal $\boldsymbol{w}$
  11. Extensions
  12. Screening Units When They are Too Many
  13. Incorporating auxiliary covariates
  14. Empirical Studies
  15. Simulation Studies
  16. ...and 4 more sections

Key Result

Theorem 1

Under Model model-po, assume that (1) $\max_t\mathbb{E} \left[\epsilon_t^4\right]\leq c_1<\infty$ for some constant $c_1$, (2) $\|\boldsymbol{\mu}_1^{(o)}\|^2/(T-T_0)\leq c_2<\infty$ for some constant $c_2$, (3) $J^{-1}(T-T_0)^{-1}T_0\|\boldsymbol{\mu}_1^{(o)}-\mathbf{Y}_0^{(o)}[\text{diag}(\mathbf{

Figures (1)

  • Figure 1: Study of the Basque Country. Upper: Actual and counterfactual per capita GDP of the Basque Country. Bottom: The difference of actual and counterfactual values, representing the average treatment effect on the treated (ATT).

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2