Difference-Based High-Dimensional Long-Run Covariance Matrix Estimation for Mean-shift Time Series

TL;DR

This paper introduces a difference-based method for estimating high-dimensional long-run covariance matrices in time series with changing means, improving accuracy and sparsity in complex, evolving data.

Contribution

It proposes a robust difference-based estimator combined with thresholding and tapering, with proven convergence rates under various dependence and mean variation conditions.

Findings

The method performs well in high-dimensional settings.

It effectively handles nonconstant mean structures.

Numerical experiments confirm its advantages over traditional estimators.

Abstract

We consider estimation of high-dimensional long-run covariance matrices for time series with nonconstant means, a setting in which conventional estimators can be severely biased. To address this difficulty, we propose a difference-based initial estimator that is robust to a broad class of mean variations, and combine it with hard thresholding, soft thresholding, and tapering to obtain sparse long-run covariance estimators for high-dimensional data. We derive convergence rates for the resulting estimators under general temporal dependence and time-varying mean structures, showing explicitly how the rates depend on covariance sparsity, mean variation, dimension, and sample size. Numerical experiments show that the proposed methods perform favorably in high dimensions, especially when the mean evolves over time.