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.

翻译：非线性动态波动性已在许多金融时间序列中被观测到。最近提出的分位数周期图提供了一种在频域中检验该现象的新方法。分位数周期图通过在不同频率和分位数水平下对时间序列数据进行三角分位数回归构建，从而实现对非线性序列依赖性的分位数-频率分析（QFA）。本文引入了一些基于分位数周期图的谱测度，用于金融时间序列模型的诊断检验以及基于模型的判别分析。采用基于模拟的参数自助法来计算这些谱测度的$p$值。通过模拟研究和一项应用实例——结合GARCH类模型分析标准普尔500指数的日对数收益率——验证了所提方法的实用性。结果表明，QFA方法能够为这些金融时间序列模型的拟合优度提供传统检验可能遗漏的额外洞见。同时，QFA方法还为检测金融时间序列中的状态转换提供了信息更丰富的判别分析途径。