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Generalized Encouragement-Based Instrumental Variables for Counterfactual Regression

Anpeng Wu, Kun Kuang, Ruoxuan Xiong, Xiangwei Chen, Zexu Sun, Fei Wu, Kun Zhang

TL;DR

This work tackles causal effect estimation under nonrandom encouragement designs with continuous treatments by introducing EnCounteR, a generalized instrumental-variables framework that integrates observational data and varied encouragements. The authors derive identifiability results for linear and nonlinear settings, recast the estimation as a GMM problem, and augment it with covariate balance and multiple moment constraints learned via adversarial representations. The method combines reweighting, moment constraints, and counterfactual regression to achieve lower-variance, more accurate CATE estimation, demonstrated across synthetic linear simulations and non-linear, real-world-like datasets (IHDP, ACIC). Empirical results show EnCounteR consistently outperforms both IV-based and covariate-based baselines, with notable robustness to non-linearities and scalability across different numbers of encouragements and data volumes. This approach enables cost-effective, high-quality causal analysis in settings where randomized encouragements are imperfect or costly, with practical impact for education, healthcare, and policy evaluation.

Abstract

In causal inference, encouragement designs (EDs) are widely used to analyze causal effects, when randomized controlled trials (RCTs) are impractical or compliance to treatment cannot be perfectly enforced. Unlike RCTs, which directly allocate treatments, EDs randomly assign encouragement policies that positively motivate individuals to engage in a specific treatment. These random encouragements act as instrumental variables (IVs), facilitating the identification of causal effects through leveraging exogenous perturbations in discrete treatment scenarios. However, real-world applications of encouragement designs often face challenges such as incomplete randomization, limited experimental data, and significantly fewer encouragements compared to treatments, hindering precise causal effect estimation. To address this, this paper introduces novel theories and algorithms for identifying the Conditional Average Treatment Effect (CATE) using variations in encouragement. Further, by leveraging both observational and encouragement data, we propose a generalized IV estimator, named Encouragement-based Counterfactual Regression (EnCounteR), to effectively estimate the causal effects. Extensive experiments on both synthetic and real-world datasets demonstrate the superiority of EnCounteR over existing methods.

Generalized Encouragement-Based Instrumental Variables for Counterfactual Regression

TL;DR

This work tackles causal effect estimation under nonrandom encouragement designs with continuous treatments by introducing EnCounteR, a generalized instrumental-variables framework that integrates observational data and varied encouragements. The authors derive identifiability results for linear and nonlinear settings, recast the estimation as a GMM problem, and augment it with covariate balance and multiple moment constraints learned via adversarial representations. The method combines reweighting, moment constraints, and counterfactual regression to achieve lower-variance, more accurate CATE estimation, demonstrated across synthetic linear simulations and non-linear, real-world-like datasets (IHDP, ACIC). Empirical results show EnCounteR consistently outperforms both IV-based and covariate-based baselines, with notable robustness to non-linearities and scalability across different numbers of encouragements and data volumes. This approach enables cost-effective, high-quality causal analysis in settings where randomized encouragements are imperfect or costly, with practical impact for education, healthcare, and policy evaluation.

Abstract

In causal inference, encouragement designs (EDs) are widely used to analyze causal effects, when randomized controlled trials (RCTs) are impractical or compliance to treatment cannot be perfectly enforced. Unlike RCTs, which directly allocate treatments, EDs randomly assign encouragement policies that positively motivate individuals to engage in a specific treatment. These random encouragements act as instrumental variables (IVs), facilitating the identification of causal effects through leveraging exogenous perturbations in discrete treatment scenarios. However, real-world applications of encouragement designs often face challenges such as incomplete randomization, limited experimental data, and significantly fewer encouragements compared to treatments, hindering precise causal effect estimation. To address this, this paper introduces novel theories and algorithms for identifying the Conditional Average Treatment Effect (CATE) using variations in encouragement. Further, by leveraging both observational and encouragement data, we propose a generalized IV estimator, named Encouragement-based Counterfactual Regression (EnCounteR), to effectively estimate the causal effects. Extensive experiments on both synthetic and real-world datasets demonstrate the superiority of EnCounteR over existing methods.
Paper Structure (27 sections, 5 theorems, 24 equations, 4 figures, 4 tables)

This paper contains 27 sections, 5 theorems, 24 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Problem Setup and Solutions
  4. Notations
  5. Assumptions and Theorems
  6. Formalization in Linear Setting
  7. GMM Reformulation
  8. Generalization to non-linear settings
  9. Methodology
  10. Reweighting for Covariate Balance
  11. Moment Constraint Learning
  12. Counterfactual Regression
  13. Implementation
  14. Empiracal Experiments
  15. Baselines and Metrics
  16. ...and 12 more sections

Key Result

Theorem 1

Under Assumptions ass:linear & ass:indep, given two datasets $\{ \mathcal{D}^{(e_0)}, \mathcal{D}^{(e_1)}\}$ with different encouragements $\{e_0,e_1\} \in \mathcal{E}$, the causal effect $\psi_t$ is identifiable.

Figures (4)

  • Figure 1: Overview of the Encouragement Design Framework. For example, in online course platforms like Coursera, edX, and Udacity, using only observational data to control observed confounders $X^{(0)}$, we can not consistently estimate the causal effects of forum engagement duration $T^{(0)}$ on exam scores $Y^{(0)}$ due to the presence of unmeasured confounders $U^{(0)}$. Therefore, we use varied encouragement policies (Class A: ${e}_\text{A}=\text{None}$, Class B: ${e}_\text{B}=\text{Praise}$, Class C: ${e}_\text{C}=\text{Points}$) to encourage longer forum engagement duration (treatments $T^{(e)}$), while these policies do not have a direct effect on exam scores (outcomes $Y^{(e)}$), which offers opportunities to identify causal effects.
  • Figure 2: Results ($\varepsilon_{\text{CE}}$) of LAS, GMM, and Our EnCounteR in Linear Simulations, with varying sample sizes $n_0 \in \{500,1000,2000,5000\}$ for observational dataset $\mathcal{D}^{(e_0)}$ and varying sample sizes $n_1 = n_0 \times \rho$ with $\rho = \{10\%,20\%,30\%,50\%,100\%\}$ for encouragement experiments $\mathcal{D}^{(e_1)}$ across various dimensions $\text{d}_\text{x} = \{2,5,10\}$ of $X$.
  • Figure 3: Box Plot of $\epsilon_{\text{PEHE}}$ in Simulation (Mults): Explore Varying Encouragements $K$ and Varying Data Volume $K \times n_k$.
  • Figure 4: Hyper-Parameter Optimization: The minimum regression error on the validation data implies the optimal hyper-parameters. The optimal hyper-parameters are $\text{d}_\text{h}=32, \text{d}_\text{r}=5, \alpha=10$ for Mult.

Theorems & Definitions (7)

  • Theorem 1
  • proof
  • Lemma 1
  • Theorem 2
  • Theorem 3
  • Corollary 1
  • proof