Vector wavelet coherence for multiple time series

Abstract

This paper introduces a new wavelet methodology to handle dynamic co-movements of multivariate time series via extending multiple and quadruple wavelet coherence methodologies. The primary motivation of our works is to measure wavelet coherence analytically for the specific dimension. Thanks to the analytical solution, both smoothed complex wavelet coherence ($C^d$) and vector wavelet coherence (${VR}^2$) can be calculated for any dimensions. Two illustrative cases employ to explore the method. The first illustration designates that dynamic co-movement between VIX and stock indices over the period between January 2000 to November 2019. Vector wavelet coherence methodology employed to examine the structure of dynamic relationships. Empirical results revealed that the relationships detected are not significant for most time-frequencies. The second application aims to approve existing multiple, quadruple, five, and six wavelet coherences. In order to validate, we generate synthetic sine curves and employ vector wavelet coherence for both multivariate (y, $x_1$, $x_2$), quadruple (y, $x_1$, $x_2$, $x_3$), five (y, $x_1$, $x_2$, $x_3$, $x_4$), and six (y, $x_1$, $x_2$, $x_3$, $x_4$, $x_5$) coherencies. The results of VMC only capture the relational frequencies and verify the existing methodologies.