Instrumental Variable Analysis Without Structural Equations

Technical Summary: Instrumental Variable Analysis Without Structural Equations

Title: Instrumental Variable Analysis Without Structural Equations
Authors: Zikai Shen, Dimitri Meunier, Houssam Zenati, Arthur Gretton, Nathan Kallus, and Aurélien Bibaut (April 2026)

1. The Problem: The Fragility of Structural Assumptions

Traditional Instrumental Variable (IV) analysis relies on the assumption that a structural equation exists: $Y = h_{\text{structural}}(X) + \epsilon$, where $X$ is endogenous. Statistical inference is typically conducted by finding a solution $h$ to the linear equation $T_P h = r_P$, where $T_P$ is an operator related to the instrument $Z$.

The authors identify two fundamental vulnerabilities in this approach:

1. Non-existence of Solutions: In real-world data, the exclusion restriction or additivity might not hold perfectly. If the statistical model assumes a perfect solution exists (as many Nonparametric IV methods do), the estimator can "deteriorate ungracefully" when no exact solution is present.
2. Identification/Uniqueness Issues: Even if solutions exist, they may not be unique (instrument weakness). If the instrument $Z$ does not sufficiently characterize the aspects of $h_{\text{structural}}$ of interest, different solutions may yield different values for the causal estimand.

2. Methodology: Targeting a Robust Estimand

To overcome these issues, the authors propose a shift in target. Instead of targeting a solution to an equation that may not have one, they target a well-defined statistical quantity $\Psi(P)$ that coincides with the causal effect when a solution exists but remains valid even when it does not.

The Proposed Estimand:
$$\Psi(P) = \mathbb{E}_P [g(Z)h(X)]$$
where $h$ and $g$ are defined as the least-squares minimizers of the primary and dual equations:

• $h \in \text{arg min}_{h \in H} \|T_P h - r_P\|^2$
• $g \in \text{arg min}_{g \in G} \|T^*_P g - a_P\|^2$

Key Methodological Innovations:

• Parametrization Invariance: They prove (Lemma 1) that while $h$ and $g$ might not be unique, the inner product $\mathbb{E}_P[g(Z)h(X)]$ is invariant to the choice of minimizer.
• Weak Riesz Representers: They introduce "weak-norm Riesz representers" ($\alpha_h, \alpha_g$) to handle the lack of exact solutions. This allows them to expand the functional bias without requiring the existence of solutions to the primary or dual equations.
• Debiased Score Function: They construct a complex score function $\chi(W; \eta)$ involving multiple nuisance functions (primary/secondary nuisances and their projections) to ensure the estimator is "doubly robust" against the errors introduced by using least-squares approximations rather than exact solutions.

3. Key Contributions and Results

The paper provides a complete framework for the estimation and inference of $\Psi(P)$ using machine learning.

• Bias Characterization (Theorem 1): They provide a detailed expansion of the bias. Unlike previous works where bias only involves the target functions, their expansion includes terms accounting for the "projection error" (the distance between the least-squares solution and the true Riesz representer).
• Nuisance Learner Framework: They develop specific minimax learners for the various components:
  • Tikhonov-regularized learners for the minimum-norm solutions $h^\dagger$ and $g^\dagger$.
  • Riesz regression learners for the representers $r_P$ and $a_P$.
  • Projection learners for the auxiliary nuisance functions $\xi$.
• Asymptotic Linearity (Theorem 6): The central result proves that the proposed estimator $\hat{\Psi}$ is $\sqrt{n}$-asymptotically linear:
  $$\sqrt{n}(\hat{\Psi} - \Psi(P)) \rightsquigarrow N(0, \sigma^2_P)$$
  This means that even with complex, non-parametric machine learning models, one can perform valid frequentist inference (constructing confidence intervals) as long as the nuisance learners converge at sufficient rates.
• Sufficient Convergence Rates: They specify the "rate requirements" for the machine learning models. For the balanced case, they show that if the critical radius $\delta_n$ of the function classes satisfies $\sqrt{n} \delta_n^{3/2} \ell_n \to 0$, the inference remains valid.

4. Significance

This paper represents a significant advancement in Robust Causal Inference.

1. Theoretical Robustness: It moves the field away from "existence-based" identification to "least-squares-based" identification. This makes the theory much more applicable to real-world data where structural assumptions are almost always slightly violated.
2. Generality: The framework is not limited to IV; it generalizes to related problems like proximal causal inference and missing-not-at-random (MNAR) data.
3. Practicality for ML: By providing explicit convergence rates and a debiased score, the paper provides a "recipe" for practitioners to use high-dimensional, non-parametric machine learning models (like neural networks or random forests) to perform causal inference while maintaining valid statistical uncertainty quantification.