Possibilistic Instrumental Variable Regression (2511.16029v1)

Abstract: Instrumental variable regression is a common approach for causal inference in the presence of unobserved confounding. However, identifying valid instruments is often difficult in practice. In this paper, we propose a novel method based on possibility theory that performs posterior inference on the treatment effect, conditional on a user-specified set of potential violations of the exogeneity assumption. Our method can provide informative results even when only a single, potentially invalid, instrument is available, offering a natural and principled framework for sensitivity analysis. Simulation experiments and a real-data application indicate strong performance of the proposed approach.

• The paper introduces a possibility-theoretic framework that models epistemic uncertainty about instrument exogeneity, enabling valid causal inference.
• It replaces integration with maximization to yield robust interval estimation even when instruments may be invalid.
• Simulation and real-world studies demonstrate reliable confidence set coverage and computational efficiency in estimating treatment effects.

Possibilistic Instrumental Variable Regression: A Possibility-Theoretic Approach to Instrumental Variables

Introduction and Motivation

Instrumental Variable (IV) regression is central to causal inference when unobserved confounding makes classical regression approaches inconsistent. Standard IV analysis crucially depends on the assumption that the chosen instruments are both relevant and valid—i.e., associated with the treatment, uncorrelated with unobserved confounders, and affecting the outcome exclusively through the treatment. In practice, these assumptions are often questionable; the exogeneity (validity) of instruments can rarely be guaranteed, which undermines the credibility of inference drawn from such models.

Building on these limitations, "Possibilistic Instrumental Variable Regression" (2511.16029) develops a novel inferential methodology grounded in possibility theory, directly modeling epistemic uncertainty about instrument exogeneity and enabling valid inference—even under single or multiple, potentially invalid, instruments.

Methodological Framework

Possibility Theory and Posterior Possibility

The methodological innovation lies in replacing the conventional probabilistic representation of uncertainty with a possibilistic one, where uncertainty is quantified via a possibility function (a supremum-normalized function assigning plausibility in $[0,1]$). This framework distinguishes itself from traditional Bayesian inference by:

• Using maximization (supremum) rather than integration in marginalization and conditioning.
• Directly accommodating vacuous (uninformative) priors, yielding robust inference even under severe identification problems.
• Avoiding the need for change-of-measure corrections (e.g., Jacobians) in parameter transformations.

Given data $Y$ from a parametric model $p(Y|\theta)$ and prior possibility $f_{\boldsymbol{\theta}}(\cdot)$, the posterior possibility is:

$$f_{\boldsymbol{\theta}|Y}(\theta | Y) = \frac{p(Y|\theta) f_{\boldsymbol{\theta}}(\theta)}{ \sup_{\theta'} p(Y| \theta') f_{\boldsymbol{\theta}}(\theta')}$$

This formulation allows for direct propagation of reduced-form estimation uncertainty to structural parameters, which is crucial when instrument validity is uncertain.

Structural Model and Identification Under Violations

Consider the standard structural system:

$$\begin{align} Y_i &= \beta X_i + Z_i \alpha + \epsilon_i \ X_i &= Z_i \gamma_2 + \eta_i \end{align}$$

where $Y_i$ is the outcome, $X_i$ the endogenous regressor, $Z_i$ the $p$-dimensional instrument vector, and both errors are jointly Gaussian. Here, $\beta$ is the treatment effect, and $\alpha$ measures direct effects of $Z$ on $Y$—violations of exogeneity correspond to nonzero $\alpha$.

Identification is obstructed if $\alpha$ is not known to be zero: the mapping from reduced-form to structural parameters is no longer bijective, and classical point identification fails.

Possibilistic Inference and Sensitivity Analysis

The central object is the posterior possibility of $\beta$—the treatment effect—given that the direct effects $\alpha$ lie in a user-specified violation set $A \subset \mathbb{R}^p$. The analyst specifies $A$ to encode prior beliefs or sensitivity analysis scenarios regarding the magnitude of IV violations.

Core Algorithmic Steps:

1. Reduced-Form Estimation: Obtain MLEs $\hat{\gamma}_1, \hat{\gamma}_2$, and covariance $\hat{\Psi}$.
2. Posterior Possibility Mapping: For a candidate value $\beta$:
   • Impute the optimal $\alpha$ as either $t(\beta) = \hat{\gamma}_1 - \beta \hat{\gamma}_2$ (if in $A$) or its projection onto $A$.
   • Compute the corresponding structural covariance and maximize the reduced-form posterior possibility under these constraints.
3. Validification: To control frequentist error rates, the possibility function is "validified" via an outer probability, yielding level sets for confidence intervals with correct coverage.

The method's posterior possibility for $\beta$ reflects the partial identification region: if $A$ is large (many violations possible), the possibility function is uninformative; as $A$ shrinks, regions of $\beta$ compatible with the data and the constraint $A$ emerge with nonzero possibility.

Practical Implication:

• Interval Estimation: Focus is shifted from point estimation to interval estimation—every value of $\beta$ compatible with some $\alpha \in A$ is equally plausible; the width of $A$ controls conservativeness.

Numerical Evaluation

Simulation Studies

Simulations with varying degrees of IV invalidity ($\alpha$) and different violation sets $A$ demonstrate:

• Coverage: The method achieves empirical confidence interval coverage that is at or above the nominal level if $A$ includes the true $\alpha$. Overly wide $A$ makes intervals conservative; too narrow $A$ yields undercoverage when violations exist.
• Comparison with Alternatives: Coverage of traditional Two-Stage Least Squares (TSLS) and Bayesian quasi-moment-based competitors sharply declines as IV violations increase, unless broad priors on violation magnitude are imposed.

Real-World Application: Institutions and Economic Output

The paper applies the method to a canonical dataset on the effect of institutional quality on country-level GDP, using settler mortality as an instrument. By varying the allowed violation region $A$ (e.g., absolute direct effects bounded by 0.1 or 0.4), the approach reveals:

• Confidence sets widen with increased $A$, yet substantive conclusions (positive effect of institutions) remain robust unless large violations are tolerated.
• The partial identification region is clearly visualized, and possibility contours for $\beta$ degrade gracefully as exogeneity is relaxed.

  Figure 1: Validified posterior possibility functions for $\beta$ (the effect of institutions) under perfectly valid instruments and under exogeneity violations of differing magnitude. The level sets above the dashed line form the $95\%$ uncertainty interval.

Theoretical Properties and Implications

• Frequentist Validity: The validified possibility posterior yields confidence sets with correct Type-I error under the assumption that the true violations are within $A$—a nontrivial result in partially identified models.
• Robustness: The inference is immune to violation sets being too large in the sense of not producing misleadingly precise intervals, in contrast to Bayesian-only methods which can be sensitive to prior misspecification [martin_valid_2023].
• Nonrestriction: In contrast to approaches making assumptions such as the majority/plurality rule [kang_instrumental_2016], the method accommodates arbitrary patterns of invalidity, supporting single-instrument designs.
• Computational Efficiency: Optimization in possibility theory is analytically tractable and does not necessitate integration or simulation, except where validification requires Monte Carlo.

Relation to Existing Literature

The approach is distinguished from partial identification [watson_bounding_2024], quasi-Bayesian [chernozhukov_plausible_2025], and Bayesian Empirical Likelihood [chib_bayesian_2018] methods:

• Existing probabilistic methods model uncertainty about instrument validity via prior distributions, often inducing very conservative posteriors for large violation budgets.
• Possibilistic IV regression provides a nonprobabilistic epistemic uncertainty modeling, naturally forming interval-valued confidence statements directly from lack of identification.
• The method prioritizes user-driven sensitivity analysis: the specification of $A$ allows a systematic examination of robustness to instrument invalidity, closely related to contemporary sensitivity modeling in causal inference [cinelli_omitted_2025, baitairian_sensitivity_2025].

Limitations and Prospects for Future Work

• Choice of Violation Set $A$: The informativeness of inference depends on how tightly $A$ can be specified, raising further questions about elicitation from domain knowledge and empirical calibration.
• Scalability and High-Dimensional IVs: For large $p$, regularization via informative possibilistic priors or computational relaxations [cella_computationally_2025] may be required.
• Nonlinear and Nonparametric Extensions: Immediate extensions of the methodology to nonparametric structural equations (see [newey_instrumental_2003], [hall_nonparametric_2005], and [singh_kernel_2019]) may require further development, especially for infinite-dimensional parameter spaces.

Conclusion

Possibilistic Instrumental Variable Regression marks a significant conceptual advancement in causal inference under instrument uncertainty. By recasting inference in possibility theory, the method enables valid, interpretable, and computationally efficient sensitivity analysis even when all instruments are potentially invalid. Practical applications and simulation evidence confirm its utility for robust causal effect estimation—so long as the scope of instrument invalidity can be circumscribed with domain knowledge. This framework complements emerging probabilistic and imprecise probability approaches [martin_possibilistic_2025], and sets the stage for broader integration of non-additive uncertainty quantification in AI and econometric methodology.