Quantile-Frequency Analysis and Spectral Measures for Diagnostic Checks of Time Series With Nonlinear Dynamics

Abstract

Nonlinear dynamic volatility has been observed in many financial time series. The recently proposed quantile periodogram offers an alternative way to examine this phenomena in the frequency domain. The quantile periodogram is constructed from trigonometric quantile regression of time series data at different frequencies and quantile levels, enabling the quantile‐frequency analysis (QFA) of nonlinear serial dependence. This paper introduces some spectral measures based on the quantile periodogram for diagnostic checks of financial time series models and for model‐based discriminant analysis. A simulation‐based parametric bootstrapping technique is employed to compute the p‐values of the spectral measures. The usefulness of the proposed method is demonstrated by a simulation study and a motivating application using the daily log returns of the S&P 500 index together with GARCH‐type models. The results show that the QFA method is able to provide additional insights into the goodness of fit of these financial time series models that may have been missed by conventional tests. The results also show that the QFA method offers a more informative way of discriminant analysis for detecting regime changes in financial time series.

The integrated copula spectrum

“…To ensure the existence of f, it suffices to assume that the γ U k (τ 1 , τ 2 ) are absolutely summable over k ∈ Z for each pair (τ 1 , τ 2 ), which we throughout implicitly assume. As shown in Dette et al (2015); Kley et al (2016a); Birr et al (2019), copula spectral densities enjoy many attractive properties; see also Li (2013Li ( , 2021 for similar findings in the setting of Laplace spectra. They exist without any moment assumptions, are invariant under strictly increasing marginal transformations (hence are scale-free), and provide a complete characterization of the pairwise copulas-hence the pairwise dependencies-of the series at arbitrary lags.…”

“…Assessing symmetry of tail dynamics. It is well known that financial time series exhibit asymmetric dependence structures in left-and right-hand tails, respectivelysee Jondeau and Rockinger (2003), Li (2021), among many others. Copula spectral distributions provide a natural model-free way to access this kind of asymmetry in tail dynamics.…”

“…An analysis of related concepts under long-range dependence can be found in Lim and Oh (2021). The utility of quantile and copula spectra for model building and model assessment is demonstrated in Birr et al (2019) and Li (2021), while an application of quantile-and copula-based spectral techniques to the analysis of cryptocurrency returns can be found in Su et al (2021). An extension of these concepts to the analysis of extreme events, which is related in spirit but different in many other respects, was considered by Davis et al (2013).…”

From zero crossings to quantile‐frequency analysis of time series with an application to nondestructive evaluation

“…The quantile periodogram directly measures the impact of the resulting trigonometric fit as an oscillatory component of the time series around the quantile. It not only services as a robust alternative to the ordinary periodogram but also possesses the capability of detecting certain nonlinear dynamics, such as time‐dependent volatilities in financial time series, 16,17 which cannot be detected by a simple application of the ordinary periodogram.…”

“…It is also different from the composite quantile periodogram proposed in the work of Lim and Oh 27 which combines quantile regressions at multiple quantile levels into a single objective. The QFA method, coupled with spectral divergence metrics, is shown in the work of Li 17 to provide additional information for diagnostic checks of financial time series models to supplement standard goodness‐of‐fit tests. For the problem of time series classification, which is the focus of this paper, the QFA method can be used in two ways.…”