In multivariate time series analysis, the 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. Although quantile coherence is a more powerful tool, its estimation remains challenging due to the high level of noise. 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 the vector autoregressive (VAR) model as an approximation to the quantile spectral matrix, along with nonparametric smoothing across quantiles. For each fixed quantile level, we obtain the VAR parameters from the quantile periodograms, then, using the Durbin-Levinson algorithm, we calculate the preliminary estimate of quantile coherence using the VAR parameters. 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 accurate 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，相干性度量了两个时间序列在不同频率上的线性依赖关系。然而，实际数据应用往往表现出频域中的非线性依赖。传统的相干性分析无法捕捉此类依赖。相比之下，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，从而刻画了非线性依赖。尽管分位数相干性是一种更强大的工具，但由于噪声水平较高，其估计仍然具有挑战性。本文引入了一种新的分位数相干性估计技术。所提出的方法为半参数方法，其利用向量自回归（VAR）模型谱的参数形式作为分位数谱矩阵的近似，并结合分位数间的非参数平滑。对于每个固定分位数水平，我们通过分位数周期图获得VAR参数，然后使用Durbin-Levinson算法，根据VAR参数计算分位数相干性的初步估计。最后，我们使用非参数平滑器对跨分位数的分位数相干性初步估计进行平滑。数值结果表明，所提出的估计方法优于非参数方法。我们证明，基于分位数相干性的二元时间序列聚类相较于普通VAR相干性具有优势。在应用中，通过与市场基准比较，基于分位数相干性识别的金融股票聚类显示出更具趣味性和更准确的多元化投资组合结构，投资者可据此做出更优决策。