Instrumental Variable Analysis Without Structural Equations

Abstract: We consider debiased inference on least-squares solutions to inverse problems as a way to avoid having to assume exact solutions exist. Such assumptions are substantive and not innocuous and their failure may well imperil inference when we impose them on the statistical model. Our approach instead allows us to conduct inference on a quantity that is defined regardless of solutions existing and coincides with the usual estimands when they do. For the case of instrumental variables, this means we can motivate the analysis with structural models but these do not need to hold exactly for the inferential procedure to remain valid.

• The paper presents a novel IV method that replaces strict structural equation assumptions with minimum-norm least-squares solutions for causal inference.
• It introduces a debiased estimator and explicit bias expansion to handle weak instruments and violations of traditional exclusion restrictions.
• The approach guarantees √n asymptotic linearity and leverages minimax and machine learning techniques for robust nuisance estimation.

Instrumental Variable Analysis Without Structural Equations

Motivation and Problem Statement

Traditional instrumental variable (IV) methods in causal inference rely on structural equations that assert the existence of an exact solution characterizing the causal effect of endogenous treatment $X$ on outcome $Y$. This equation, typically $Y = h_{\text{structural}}(X) + \epsilon$, assumes $h_{\text{structural}}$ exists within a specified function class and imposes strong requirements on exclusion and homogeneity. Nonparametric IV approaches relax parametric constraints, but still maintain the assumption that a solution to the crucial conditional moment restriction (CMR) equation exists.

However, these existence assumptions are far from innocuous. When violated—be it through weak instruments, exclusion restriction failures, or heterogeneous noise—the inferential procedures deteriorate ungracefully. The paper proposes a framework where such structural assumptions can be motivating, but are not required for valid statistical inference.

Estimand Without Structural Solutions

Instead of targeting quantities defined only under exact solvability of CMRs, the authors introduce inference on minimum-norm least-squares solutions to inverse problems. The estimand $\Psi(P)$ is defined using minimizers of squared residuals: $$\Psi(P) = \mathbb{E}_P[g(Z) h(X)], \quad h \in \mathcal{H}_P, \quad g \in \mathcal{G}_P,$$ where $\mathcal{H}_P$ and $\mathcal{G}_P$ are affine sets of least-squares minimizers for the forward and dual equations, respectively. Critically, these sets are guaranteed non-empty under a mild "range condition" on the population Riesz representers, without requiring existence of exact solutions.

When exact structural solutions exist, this construction coincides with conventional causal estimands. Otherwise, it yields well-defined inference that degrades gracefully under mild violations.

Identification and Parametrization Invariance

The paper formalizes the existence of least-squares solutions via orthogonal decompositions for the relevant Hilbert spaces and operators. Under the range condition, the minimum-norm solutions can be interpreted via pseudoinverse or as a Tikhonov-regularized limit. The parametrization invariance lemma proves that any pair of least-squares minimizers $(h, g)$ yields the same value for $\mathbb{E}_P[h(X)g(Z)]$, ensuring that the estimand is independent of specific minimizer selection.

Debiased Estimation and Bias Expansion

Estimation is achieved through the construction of a debiased estimator: $Y$0 where $Y$1 is a tailored score function involving primary and secondary nuisances and various projections. The bias of the estimator admits an explicit expansion, including terms corresponding to mixed products of estimation errors in primary and dual functions. Notably, additional nuisances ($Y$2 and $Y$3) must be estimated to account for the lack of structural solutions, yielding extra terms in the bias decomposition not present in classical settings.

A strong claim is that when exact structural solutions are available, all non-classical bias terms vanish, reducing the procedure to standard doubly-robust IV estimation.

Minimax and Least-Squares Learning of Nuisances

The authors construct minimax learning algorithms for the minimum-norm least-squares estimators ($Y$4 and $Y$5), demonstrate fast rates under source conditions, and show strong/weak norm convergence. Analogous minimax objectives are used to estimate weak-norm Riesz representers ($Y$6, $Y$7), and standard nonparametric least-squares learners are applied to projections ($Y$8). Automatic debiased machine learning via Riesz regression is deployed for strong-norm representers ($Y$9, $Y = h_{\text{structural}}(X) + \epsilon$0), with concentration results guaranteeing $Y = h_{\text{structural}}(X) + \epsilon$1 convergence at rates determined by localized Rademacher complexities.

Asymptotic Theory and Statistical Guarantees

Under consistency and product rate conditions on the nuisance errors, the estimator $Y = h_{\text{structural}}(X) + \epsilon$2 is shown to be $Y = h_{\text{structural}}(X) + \epsilon$3-asymptotically linear and normal, with influence function determined by the debiased score. Plug-in learners achieve the necessary rates for both strong and weak norms. The expansion and analysis explicitly characterize how the estimator degrades in the presence of instrument weakness or approximate exclusion, rather than failing abruptly.

Practical and Theoretical Implications

Practically, this framework allows principled inference in settings where structural solutions are only approximately valid. This is advantageous in empirical economics, health, and other scientific domains where exact instrumental restrictions rarely hold. Theoretically, the paper generalizes efficient influence function theory for inverse problems and recasts identification in terms of weak norm projections and source conditions, broadening applicable statistical models.

The explicit bias characterization enables robust estimation and inference in the presence of instrument weakness, non-uniqueness, or ill-posed inverse equations. The construction admits flexible nuisance estimation, including plug-in machine learning, and yields valid asymptotics without requiring exact structural identification.

Conclusion

This work establishes a statistical theory of instrumental variable analysis independent of structural equations, formalizing inference via minimum-norm least-squares minimizers and explicitly debiasing for violations of structural assumptions. The framework jointly advances practical IV methodology and theoretical understanding of non-exact identification, and enables robust estimation in complex settings with weak instruments or non-homogeneous residuals. Future developments may focus on adaptive nuisance learning, relaxations for high-dimensional covariates, and extending this framework to multi-stage or longitudinal causal inference scenarios (2604.24660).