Multivariate Time Series Analysis and Applications

9Multivariate spectral analysis of time series

Similar to the univariate time series analysis, where one can study a univariate time series through its autocovariance/autocorrelation functions and lag relationships or through its spectrum properties, we can study a multivariate time series through a time domain approach or a frequency domain approach. In the time domain approach, we use the covariance/correlation matrices, and in frequency domain approach we will use the spectrum matrices. In this chapter, after a brief review of the univariate frequency domain method, we will introduce the spectral analysis for both stationary and nonstationary vector time series. With no loss of generality, we will assume a zero‐mean time series in the following discussion.

9.1 Introduction

Recall that for a univariate stationary time series process, Z_t, its spectral representation is given by

(9.1)

where dU(ω) is a complex‐valued orthogonal stochastic process for each ω such that

(9.2)

(9.3)

and

(9.4)

where U^*(ω) is the complex conjugate of U(ω). Let γ_k be the autocovariance function of