  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. It is a useful tool for quantile-frequency
analysis (QFA) of nonlinear serial dependence. This paper introduces a number
of spectral divergence metrics based on the quantile periodogram for diagnostic
checks of financial time series models and model-based discriminant analysis.
The parametric bootstrapping technique is employed to compute the $p$-values of
the metrics. The usefulness of the proposed method is demonstrated empirically
by a case study using the daily log returns of the S\&P 500 index over three
periods of time together with their 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 time series.
