The quantile spectral density and comparison based tests for nonlinear time series

TL;DR

This paper introduces tests for nonlinear time series based on the quantile spectral density, providing a new way to measure serial dependence and compare time series under weak conditions.

Contribution

It proposes a novel estimator for the quantile spectral density and develops goodness of fit and comparison tests for nonlinear time series.

Findings

Estimator for quantile spectral density shown to have good asymptotic properties

Tests effectively detect differences in dependence structures in simulations

Application to real data demonstrates practical utility

Abstract

In this paper we consider tests for nonlinear time series, which are motivated by the notion of serial dependence. The proposed tests are based on comparisons with the quantile spectral density, which can be considered as a quantile version of the usual spectral density function. The quantile spectral density 'measures' sequential dependence structure of a time series, and is well defined under relatively weak mixing conditions. We propose an estimator for the quantile spectral density and derive its asympototic sampling properties. We use the quantile spectral density to construct a goodness of fit test for time series and explain how this test can also be used for comparing the sequential dependence structure of two time series. The method is illustrated with simulations and some real data examples.