Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Possibilistic Instrumental Variable Regression

Gregor Steiner, Jeremie Houssineau, Mark F. J. Steel

TL;DR

This paper tackles causal inference with instrumental variables under uncertain instrument validity by introducing a possibilistic Bayesian framework. By modeling exogeneity violations through a user-specified set $A$, it derives a conditional posterior possibility for the structural effect $\beta$ via a reduced-form to structural-parameter mapping, with closed-form solutions under vacuous priors and a validified posterior to control type-I error. The approach yields informative, partially identified inference when $A$ is reasonably restricted and remains always defined even if all instruments are invalid, differing from ad-hoc widening of uncertainty intervals in probabilistic methods. Empirical results from simulations and a real-data application on institutions and growth demonstrate robust positive effects under plausible violation sets, highlighting the method’s practical utility for sensitivity analysis without asserting instrument validity. Overall, the framework provides a principled, computationally tractable way to quantify epistemic uncertainty about instrument exogeneity and its impact on causal estimates.

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.

Possibilistic Instrumental Variable Regression

TL;DR

This paper tackles causal inference with instrumental variables under uncertain instrument validity by introducing a possibilistic Bayesian framework. By modeling exogeneity violations through a user-specified set , it derives a conditional posterior possibility for the structural effect via a reduced-form to structural-parameter mapping, with closed-form solutions under vacuous priors and a validified posterior to control type-I error. The approach yields informative, partially identified inference when is reasonably restricted and remains always defined even if all instruments are invalid, differing from ad-hoc widening of uncertainty intervals in probabilistic methods. Empirical results from simulations and a real-data application on institutions and growth demonstrate robust positive effects under plausible violation sets, highlighting the method’s practical utility for sensitivity analysis without asserting instrument validity. Overall, the framework provides a principled, computationally tractable way to quantify epistemic uncertainty about instrument exogeneity and its impact on causal estimates.

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.

Paper Structure

This paper contains 15 sections, 3 theorems, 38 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Possibility theory
  3. Proposed methodology
  4. The model
  5. Possibilistic inference
  6. Validification
  7. Experiments
  8. Simulation experiments
  9. A real data example: The effect of institutions on economic growth
  10. Conclusion
  11. Derivations
  12. The structural posterior possibility function
  13. Proof of Proposition \ref{['prop:conditional_posterior']} and computational considerations
  14. Proof of Proposition \ref{['prop:validity']}
  15. Additional details on the experiments

Key Result

Proposition 1

Let $A \subseteq \mathbb{R}^p$ be the considered violation set. Denote by $(\hat{\gamma}_1, \hat{\gamma}_2)$ and $\hat{\Psi}$ the maximum-likelihood estimates of the reduced-form coefficients and covariance matrix, respectively, and define $t(\beta) := \hat{\gamma}_1 - \beta \hat{\gamma}_2$. Then, t where and $\mathrm{Proj}_A^{Z^{\intercal}Z}$ denotes the projection onto $A$ with respect to the m

Figures (2)

  • Figure 1: A geometric illustration of our method for $p=2$: The causal effect $\beta$ is partially identified where the affine subspace $t(\beta) = \Hat{\gamma}_1 - \beta \Hat{\gamma}_2$ intersects the tolerated region $A$. At these values of $\beta$, the corresponding $\alpha$ is precisely $t(\beta)$, and therefore, the conditional possibility is $1$. For all other values of $\beta$, the optimal $\alpha$ is the projection onto $A$ w.r.t. the metric induced by $Z^\intercal Z$.
  • Figure 2: The effect of institutions on economic growth: Validified posterior possibility functions under a perfectly valid instrument ($\alpha = 0$) and potential violations $A = [-0.1, 0.1]$ and $A =[-0.4, 0.4]$. The solid line is based on the $\chi^2$ approximation, while the dashed line displays the Monte Carlo approximation. The dashed grey line indicates the 0.05 level, such that the 95% uncertainty interval for $\beta$ includes all values where the posterior possibility function exceeds this threshold.

Theorems & Definitions (7)

  • Remark 1
  • Remark 2
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Corollary 1