Specification tests for regression models with measurement errors

2025-11-06

AI Summary

This paper addresses the challenge of specification testing for regression models when explanatory variables are subject to measurement error. We propose a novel test based on a deconvolution residual marked empirical process. Orthogonal projection is employed to eliminate interference from parameter estimation, and the multiplier bootstrap is introduced—first in this context—for critical value approximation, accommodating both known and unknown measurement error distributions; this effectively decouples critical value simulation from estimation effects. Integrating deconvolution kernel estimation with the integrated conditional moment (ICM) framework, the method is theoretically justified and asymptotically valid. Monte Carlo simulations demonstrate that the test achieves excellent finite-sample size control and statistical power, while exhibiting robustness to misspecification of the measurement error distribution. Consequently, it substantially broadens the applicability and practical utility of specification tests in measurement error settings.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Search and Optimization: Non-convex Optimization
• Intelligent Robots: State Estimation

Application Category

• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Web measurements
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

In this paper, we propose new specification tests for regression models with measurement errors in the explanatory variables. Inspired by the integrated conditional moment (ICM) approach, we use a deconvoluted residual-marked empirical process and construct ICM-type test statistics based on it. The issue of measurement errors is addressed by applying a deconvolution kernel estimator in constructing the residuals. We demonstrate that employing an orthogonal projection onto the tangent space of nuisance parameters not only eliminates the parameter estimation effect but also facilitates the simulation of critical values via a computationally simple multiplier bootstrap procedure. It is the first time a multiplier bootstrap has been proposed in the literature of specification testing with measurement errors. We also develop specification tests and the multiplier bootstrap procedure when the measurement error distribution is unknown. The finite-sample performance of the proposed tests for both known and unknown measurement error distributions is evaluated through Monte Carlo simulations, which demonstrate their efficacy.

Problem

• Developing specification tests for regression models with measurement errors
• Addressing measurement errors using deconvolution kernel estimators
• Proposing multiplier bootstrap for critical values with nuisance parameters

Innovation

• Deconvoluted residual-marked empirical process for tests
• Orthogonal projection eliminates nuisance parameter effects
• Multiplier bootstrap enables simple critical value simulation