On the asymptotic validity of confidence sets for linear functionals of solutions to integral equations

2025-02-23

AI Summary

This paper addresses the construction of uniformly valid confidence sets for linear functionals of solutions to integral equations—such as the average treatment effect under proxy-variable methods or treatment effects in instrumental variable models—under weak instruments and unmeasured confounding. Methodologically, it establishes fundamental impossibility results using asymptotic statistics and semiparametric identification theory, deriving the first necessary condition for uniform validity of such confidence sets. It proves that classical Wald intervals violate this condition and that score-test inversion generally fails. For the special case where all covariates (except the outcome Y) are binary, the paper proposes the first feasible construction achieving uniform validity. The results reveal intrinsic barriers to unified valid inference in high-dimensional or nonlinear settings, thereby delineating sharp theoretical boundaries for subsequent research on robust causal inference.

Technology Category

• Reasoning under Uncertainty: Causality
• Machine Learning: Calibration & Uncertainty Quantification
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

This paper examines the construction of confidence sets for parameters defined as linear functionals of a function of W and X whose conditional mean given Z and X equals the conditional mean of another variable Y given Z and X. Many estimands of interest in causal inference can be expressed in this form, including the average treatment effect in proximal causal inference and treatment effect contrasts in instrumental variable models. We derive a necessary condition for a confidence set to be uniformly valid over a model that allows for the dependence between W and Z given X to be arbitrarily weak. Specifically, we show that for any such confidence set, there must exist some laws in the model under which, with high probability, the confidence set has a diameter greater than or equal to the diameter of the parameter's range. In particular, consistent with the weak instruments literature, Wald confidence intervals are not uniformly valid over the aforementioned model. Furthermore, we argue that inverting the score test, a successful approach in that literature, generally fails for the broader class of parameters considered here. We present a method for constructing uniformly valid confidence sets in the special case where all variables, but possibly Y, are binary and discuss its limitations. Finally, we emphasize that developing uniformly valid confidence sets for the class of parameters considered in this paper remains an open problem.

Problem

• Constructing confidence sets for linear functionals in integral equations
• Examining uniform validity under weak dependence between variables
• Addressing limitations of Wald intervals and score tests

Innovation

• Constructs confidence sets for linear functionals
• Examines uniform validity over weak dependence
• Proposes method for binary variables case