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Constructive Instrumental Variable Identification and Inference with Many Weak Interaction Moments

Di Zhang, Minhao Yao, Zhonghua Liu, Baoluo Sun

Abstract

Instrumental variable methods are widely used for causal inference, but identification becomes especially challenging when instruments are weak and potentially invalid. These challenges are particularly pronounced in Mendelian randomization, where genetic variants serve as instruments and violations of exclusion restriction or independence assumptions are common. We propose MAGIC, a constructive and assumption-lean framework that achieves identification even when all candidate instruments may be invalid. The method exploits pairwise and higher-order interactions among mutually independent instruments to construct moment conditions orthogonal to both unmeasured confounding and direct effects under a linear structural model. The resulting estimation problem involves many potentially weak interaction moments with unknown nuisance parameters. We develop a semiparametric generalized method of moments estimator and introduce a global Neyman orthogonality condition to ensure robustness of both the moment function and its derivative to nuisance estimation under many weak moment asymptotics. We establish consistency and asymptotic normality when the number of moments diverges with sample size and characterize the semiparametric efficiency bound under fixed dimension. Simulations and an application to UK Biobank data illustrate the method.

Constructive Instrumental Variable Identification and Inference with Many Weak Interaction Moments

Abstract

Instrumental variable methods are widely used for causal inference, but identification becomes especially challenging when instruments are weak and potentially invalid. These challenges are particularly pronounced in Mendelian randomization, where genetic variants serve as instruments and violations of exclusion restriction or independence assumptions are common. We propose MAGIC, a constructive and assumption-lean framework that achieves identification even when all candidate instruments may be invalid. The method exploits pairwise and higher-order interactions among mutually independent instruments to construct moment conditions orthogonal to both unmeasured confounding and direct effects under a linear structural model. The resulting estimation problem involves many potentially weak interaction moments with unknown nuisance parameters. We develop a semiparametric generalized method of moments estimator and introduce a global Neyman orthogonality condition to ensure robustness of both the moment function and its derivative to nuisance estimation under many weak moment asymptotics. We establish consistency and asymptotic normality when the number of moments diverges with sample size and characterize the semiparametric efficiency bound under fixed dimension. Simulations and an application to UK Biobank data illustrate the method.

Paper Structure

This paper contains 19 sections, 5 theorems, 50 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Constructive Identification
  3. Data Structure and Notation
  4. The ALICE Model with Possibly Invalid Instruments
  5. Identification via Constructed IV Interactions
  6. Semiparametric Efficiency Theory
  7. Estimation under Many Weak Interaction Moments
  8. Continuous Updating Estimation with Known Nuisance
  9. Many Weak Moments and Nuisance Estimation
  10. Global Neyman Orthogonality
  11. Orthogonalized Interaction Moments
  12. The MAGIC Estimator
  13. Theoretical Properties
  14. Many Weak Moment Asymptotics
  15. Asymptotic Results
  16. ...and 4 more sections

Key Result

Theorem 1

Suppose Assumptions assp:alice, assp:indp, and assp:relevance hold. Then the true parameter value $\beta=\beta^{\ast}$ is the unique solution to the population moment condition $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: Illustration of the many weak interaction asymptotic regime. The vertical axis indexes the rate at which interaction–exposure correlations decay, while the horizontal axis indexes the growth rate of the number of interaction moments. The shaded region shows combinations of these rates for which the effective identification strength satisfy Assumption \ref{['assp:mwi']}.

Theorems & Definitions (5)

  • Theorem 1: Constructive Identification
  • Theorem 2: Semiparametric Efficiency Bound
  • Lemma 1: Global Neyman Orthogonality
  • Theorem 3: Consistency and Asymptotic Normality
  • Theorem 4: Test of Overidentifying Restrictions