Local Quadratic Spectral and Covariance Matrix Estimation

TL;DR

This paper introduces a new local polynomial estimator for the spectral density matrix of multivariate time series at boundary frequencies, improving covariance estimation crucial for statistical inference on the mean.

Contribution

It proposes a novel boundary-specific estimator for spectral density matrices using local polynomial regression, with theoretical analysis and practical applications.

Findings

Estimator performs well at boundary frequencies in simulations

Improves covariance matrix estimation for statistical inference

Applicable to economic data like inflation and unemployment

Abstract

The problem of estimating the spectral density matrix $f(w)$ of a multivariate time series is revisited with special focus on the frequencies $w=0$ and $w=\pi$. Recognizing that the entries of the spectral density matrix at these two boundary points are real-valued, we propose a new estimator constructed from a local polynomial regression of the real portion of the multivariate periodogram. The case $w=0$ is of particular importance, since $f(0)$ is associated with the large-sample covariance matrix of the sample mean; hence, estimating $f(0)$ is crucial in order to conduct any sort of statistical inference on the mean. We explore the properties of the local polynomial estimator through theory and simulations, and discuss an application to inflation and unemployment.