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Identification and Debiased Learning of Causal Effects with General Instrumental Variables

Shuyuan Chen, Peng Zhang, Yifan Cui

TL;DR

This work addresses causal effect estimation with general instrumental variables by developing an additive IV (AIV) framework that nonparametrically identifies mean potential outcomes and the ATE under multi-categorical or continuous instruments and treatments. It leverages semiparametric theory to derive efficient influence functions and constructs debiased estimators via cross-fitting under two weighting schemes: a prespecified weight and an adaptive, data-driven weight, with both point-exposure and longitudinal extensions. The authors extend identification to longitudinal data and dynamic treatment regimes, including multiplicative IVs, and validate the approach through simulations and an empirical analysis of the Job Training Partnership Act program. The methodology delivers robust, efficient inference under flexible nuisance modeling, enabling valid causal conclusions in complex IV settings while highlighting practical considerations for weighting choices and longitudinal analysis.

Abstract

Instrumental variable methods are fundamental to causal inference when treatment assignment is confounded by unobserved variables. In this article, we develop a general nonparametric framework for identification and learning with multi-categorical or continuous instrumental variables. Specifically, we propose an additive instrumental variable framework to identify mean potential outcomes and the average treatment effect with a weighting function. Leveraging semiparametric theory, we derive efficient influence functions and construct consistent, asymptotically normal estimators via debiased machine learning. Extensions to longitudinal data, dynamic treatment regimes, and multiplicative instrumental variables are further developed. We demonstrate the proposed method by employing simulation studies and analyzing real data from the Job Training Partnership Act program.

Identification and Debiased Learning of Causal Effects with General Instrumental Variables

TL;DR

This work addresses causal effect estimation with general instrumental variables by developing an additive IV (AIV) framework that nonparametrically identifies mean potential outcomes and the ATE under multi-categorical or continuous instruments and treatments. It leverages semiparametric theory to derive efficient influence functions and constructs debiased estimators via cross-fitting under two weighting schemes: a prespecified weight and an adaptive, data-driven weight, with both point-exposure and longitudinal extensions. The authors extend identification to longitudinal data and dynamic treatment regimes, including multiplicative IVs, and validate the approach through simulations and an empirical analysis of the Job Training Partnership Act program. The methodology delivers robust, efficient inference under flexible nuisance modeling, enabling valid causal conclusions in complex IV settings while highlighting practical considerations for weighting choices and longitudinal analysis.

Abstract

Instrumental variable methods are fundamental to causal inference when treatment assignment is confounded by unobserved variables. In this article, we develop a general nonparametric framework for identification and learning with multi-categorical or continuous instrumental variables. Specifically, we propose an additive instrumental variable framework to identify mean potential outcomes and the average treatment effect with a weighting function. Leveraging semiparametric theory, we derive efficient influence functions and construct consistent, asymptotically normal estimators via debiased machine learning. Extensions to longitudinal data, dynamic treatment regimes, and multiplicative instrumental variables are further developed. We demonstrate the proposed method by employing simulation studies and analyzing real data from the Job Training Partnership Act program.
Paper Structure (52 sections, 27 theorems, 171 equations, 2 figures, 4 tables, 3 algorithms)

This paper contains 52 sections, 27 theorems, 171 equations, 2 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Outlines of our paper
  4. Identification
  5. Basic assumptions
  6. Solving nonparametric IV models
  7. Identification of average treatment effect
  8. Semiparametric theory
  9. Prespecified weighting function scenario
  10. Adaptive weighting function scenario
  11. Cross-fitting procedure
  12. Extension to longitudinal data
  13. Identification strategy
  14. Semiparametric theory
  15. Cross-fitting procedure
  16. ...and 37 more sections

Key Result

Proposition 2.1

For each $a \in \mathcal{A}$, $Z$ is an AIV for $A = a$ if and only if for any $\pi(Z,L)$, $\mathrm{Cov}\{A^{(a)},\pi(Z,L)\mid U,L\}=\mathrm{Cov}\{A^{(a)},\pi(Z,L)\mid L\}.$

Figures (2)

  • Figure 4.1: Sequential DAG when $T=2$. Gray nodes indicate unobserved variables, while white nodes indicate observed ones.
  • Figure 4.2: Illustration of the backward cross-fitting procedure in the $k$-th fold for $T=2$.

Theorems & Definitions (42)

  • Definition 2.1: Additive IV
  • Proposition 2.1
  • Definition 2.2: Regular weighting function
  • Proposition 2.2: Existence
  • Theorem 2.1: Uniqueness and closed form solution
  • Proposition 2.3: Identification of mean potential outcomes
  • Proposition 2.4: Identification of ATE
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • ...and 32 more