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Nonlinear dynamic volatility has been observed in many financial time series.The recently proposed quantile periodogram offers an alternative way to examinethis phenomena in the frequency domain. The quantile periodogram is constructedfrom trigonometric quantile regression of time series data at differentfrequencies and quantile levels. It is a useful tool for quantile-frequencyanalysis (QFA) of nonlinear serial dependence. This paper introduces a numberof spectral divergence metrics based on the quantile periodogram for diagnosticchecks of financial time series models and model-based discriminant analysis.The parametric bootstrapping technique is employed to compute the $p$-values ofthe metrics. The usefulness of the proposed method is demonstrated empiricallyby a case study using the daily log returns of the S\&P 500 index over threeperiods of time together with their GARCH-type models. The results show thatthe QFA method is able to provide additional insights into the goodness of fitof these financial time series models that may have been missed by conventionaltests. The results also show that the QFA method offers a more informative wayof discriminant analysis for detecting regime changes in time series.

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