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

TL;DR

This paper introduces a new method for constructing confidence regions for linear regression parameters under stationary and ergodic dependent errors, avoiding explicit long-run variance estimation.

Contribution

It develops a novel joint inference approach using random smoothing and data-driven bandwidth selection applicable to various dependent error structures.

Findings

Achieves near-nominal coverage in simulations with complex dependence structures.

Provides confidence regions with competitive volume compared to existing methods.

Demonstrates practical application with 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