Empirical spectral processes for locally stationary time series

TL;DR

This paper develops a new empirical spectral process framework for locally stationary time series, establishing weak convergence and inequalities without Gaussian assumptions, with applications in estimation and testing.

Contribution

It introduces a novel empirical spectral process for locally stationary series, deriving convergence results under metric entropy conditions without Gaussian assumptions.

Findings

Established weak convergence in a function space.

Proved a maximal exponential inequality and Glivenko--Cantelli-type result.

Derived uniform convergence rates for local Whittle estimates.

Abstract

A time-varying empirical spectral process indexed by classes of functions is defined for locally stationary time series. We derive weak convergence in a function space, and prove a maximal exponential inequality and a Glivenko--Cantelli-type convergence result. The results use conditions based on the metric entropy of the index class. In contrast to related earlier work, no Gaussian assumption is made. As applications, quasi-likelihood estimation, goodness-of-fit testing and inference under model misspecification are discussed. In an extended application, uniform rates of convergence are derived for local Whittle estimates of the parameter curves of locally stationary time series models.