In multivariate time series analysis, spectral coherence measures the linear dependency between two time series at different frequencies. However, real data applications often exhibit nonlinear dependency in the frequency domain. Conventional coherence analysis fails to capture such dependency. The quantile coherence, on the other hand, characterizes nonlinear dependency by defining the coherence at a set of quantile levels based on trigonometric quantile regression. This paper introduces a new estimation technique for quantile coherence. The proposed method is semi-parametric, which uses the parametric form of the spectrum of a vector autoregressive (VAR) model to approximate the quantile coherence, combined with nonparametric smoothing across quantiles. At a given quantile level, we compute the quantile autocovariance function (QACF) by performing the Fourier inverse transform of the quantile periodograms. Subsequently, we utilize the multivariate Durbin-Levinson algorithm to estimate the VAR parameters and derive the estimate of the quantile coherence. Finally, we smooth the preliminary estimate of quantile coherence across quantiles using a nonparametric smoother. Numerical results show that the proposed estimation method outperforms nonparametric methods. We show that quantile coherence-based bivariate time series clustering has advantages over the ordinary VAR coherence. For applications, the identified clusters of financial stocks by quantile coherence with a market benchmark are shown to have an intriguing and more informative structure of diversified investment portfolios that may be used by investors to make better decisions.

翻译：在多变量时间序列分析中，谱相干性衡量了两个时间序列在不同频率上的线性依赖关系。然而，实际数据应用常常表现出频域中的非线性依赖关系。传统的相干性分析无法捕捉这种依赖关系。另一方面，分位数相干性通过基于三角分位数回归在一组分位数水平上定义相干性来刻画非线性依赖关系。本文介绍了一种分位数相干性的新估计技术。所提出的方法是半参数的，它利用向量自回归（VAR）模型频谱的参数形式来逼近分位数相干性，并结合跨分位数的非参数平滑。在给定的分位数水平上，我们通过对分位数周期图进行傅里叶逆变换来计算分位数自协方差函数（QACF）。随后，我们利用多元Durbin-Levinson算法估计VAR参数并推导出分位数相干性的估计值。最后，我们使用非参数平滑器对分位数相干性的初步估计进行跨分位数平滑。数值结果表明，所提出的估计方法优于非参数方法。我们展示了基于分位数相干性的双变量时间序列聚类相较于普通VAR相干性的优势。在应用中，通过带有市场基准的分位数相干性识别出的金融股票聚类被证明具有有趣且更具信息性的多元化投资组合结构，投资者可能利用这些结构做出更好的决策。