New Confidence Regions for Linear Regression Parameters with Stationary-Ergodic Dependent Errors

2026-05-19

AI Summary

This study addresses the challenge of constructing valid joint confidence regions for linear regression coefficients when regression errors exhibit unknown serial dependence and are jointly stationary and ergodic with the covariates. The authors propose a novel approach that avoids explicit modeling of the error dependence structure by introducing independent auxiliary samples and applying stochastic smoothing with a decaying bandwidth to both the regression function and second moments. Coupled with data-driven bandwidth selection and mild truncation, this method yields Wald-type confidence regions and simultaneous confidence intervals. It does not rely on long-run variance estimation or parametric assumptions about dependence, achieving coverage probabilities close to nominal levels across diverse dependence structures—including ARMA, ARFIMA, copula-based Markov processes, and fractional Gaussian noise—while producing smaller confidence region volumes than Newey–West HAC and MAC methods. The approach is successfully demonstrated in an analysis of Beijing PM2.5 data.

Abstract

We develop joint confidence regions for linear regression coefficients when the regressors and errors are jointly stationary and ergodic with unspecified serial dependence. The method applies random smoothing, using an independent auxiliary sample and shrinking bandwidth, to a vector of regression and second-moment statistics. Under stationarity, ergodicity, and finite second moments, the estimator is asymptotically normal and yields Wald confidence regions and simultaneous confidence intervals without direct long-run variance estimation or a parametric dependence model. For implementation, we introduce a scaled estimator with data-driven bandwidth selection and a mild truncation that improves finite-sample stability. Simulations under ARMA, ARFIMA, copula-based Markov errors, and fractional Gaussian noise, with Gaussian and heavy-tailed margins, show near-nominal coverage and competitive region volumes relative to Newey-West HAC and MAC. A winter Beijing PM2.5 application illustrates the procedure. Keywords: Random smoothing, Joint inference, Confidence regions, Dependent errors, Long memory, Regression inference

Problem

• Confidence regions
• Dependent errors
• Regression inference
• Stationary-ergodic
• Joint inference

Innovation

• Random smoothing
• Joint inference
• Confidence regions
• Dependent errors
• Regression inference