Uniform Validity of the Subset Anderson-Rubin Test under Heteroskedasticity and Nonlinearity

2025-07-01

AI Summary

This paper addresses the problem of constructing subset Anderson–Rubin (AR) tests with uniform validity for nonlinear moment models under heteroskedasticity, weak identification, partial identification, and even failure of identification. We propose a novel subset AR test based on a continuous-updating estimation criterion function. By introducing a joint perturbation–regularization construction technique, we derive an upper bound on the first-order conditions, thereby relaxing the conventional reliance on homoskedasticity and strong identification. The method imposes no assumptions of time-series stationarity or cross-sectional homogeneity, and applies to nonparametric distributional settings subject to generalized moment restrictions. We establish theoretically that the proposed test achieves correct uniform asymptotic size across all identification regimes. This substantially enhances the robustness and applicability of inference in complex data structures—particularly those involving mixed cross-sectional and time-series dependence.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Search and Optimization: Non-convex Optimization
• Game Theory and Economic Paradigms: Imperfect Information

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Sustainability of Web economics
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We consider the Anderson-Rubin (AR) statistic for a general set of nonlinear moment restrictions. The statistic is based on the criterion function of the continuous updating estimator (CUE) for a subset of parameters not constrained under the Null. We treat the data distribution nonparametrically with parametric moment restrictions imposed under the Null. We show that subset tests and confidence intervals based on the AR statistic are uniformly valid over a wide range of distributions that include moment restrictions with general forms of heteroskedasticity. We show that the AR based tests have correct asymptotic size when parameters are unidentified, partially identified, weakly or strongly identified. We obtain these results by constructing an upper bound that is using a novel perturbation and regularization approach applied to the first order conditions of the CUE. Our theory applies to both cross-sections and time series data and does not assume stationarity in time series settings or homogeneity in cross-sectional settings.

Problem

• Ensures uniform validity of AR tests under heteroskedasticity and nonlinearity
• Validates subset tests for unidentified or weakly identified parameters
• Applies to cross-sections and time series without stationarity assumptions

Innovation

• Nonparametric data distribution with parametric moment restrictions
• Novel perturbation and regularization approach
• Uniform validity under heteroskedasticity and nonlinearity