Multivariate Time Series

Abstract

Many time series arising in practice are best considered as components of some vector-valued (multivariate) time series {X _t } whose specification includes not only the serial dependence of each component series {X _tj } but also the interdependence between different component series {X _ti } and {X _tj }. From a second order point of view a stationary multivariate time series is determined by its mean vector, μ = E X, and its covariance matrices Γ(h) = E(X _{t +h} X_t′) — μμ′, h = 0, ± 1,.... Most of the basic theory of univariate time series extends in a natural way to multivariate series but new problems arise. In this chapter we show how the techniques developed earlier for univariate series are extended to the multivariate case. Estimation of the basic quantities μ and Γ(·) is considered in Section 11.2. In Section 11.3 we introduce multivariate ARMA processes and develop analogues of some of the univariate results in Chapter 3. The prediction of stationary multivariate processes, and in particular of ARMA processes, is treated in Section 11.4 by means of a multivariate generalization of the innovations algorithm used in Chapter 5. This algorithm is then applied in Section 11.5 to simplify the calculation of the Gaussian likelihood of the observations {X ₁, X ₂,..., X _n } of a multivariate ARMA process. Estimation of parameters using maximum likelihood and (for autoregressive models) the Yule—Walker equations is also considered. In Section 11.6 we discuss the cross spectral density of a bivariate stationary process {X _t } and its interpretation in terms of the spectral representation of {X _t }. (The spectral representation is discussed in more detail in Section 11.8.) The bivariate periodogram and its asymptotic properties are examined in Section 11.7 and Theorem 11.7.1 gives the asymptotic joint distribution for a linear process of the periodogram matrices at frequencies λ ₁, λ ₂,..., λ _m ∈ (0, π). Smoothing of the periodogram is used to estimate the cross-spectrum and hence the cross-amplitude spectrum, phase spectrum and squared coherency for which approximate confidence intervals are given. The chapter ends with an introduction to the spectral representation of an m-variate stationary process and multivariate linear filtering.