"Frontmatter". In: Analysis of Financial Time Series

300 VECTOR TIME SERIESthe multivariate process r t . The chapter also discusses methods that can simplify thedynamic structure or reduce the dimension of r t .Many of the models and methods discussed in the previous chapters can be generalizeddirectly to the multivariate case. But there are situations in which the generalizationrequires some attention. In some situations, one needs new models andmethods to handle the complicated relationships between multiple returns. We alsodiscuss methods that search for common factors affecting the returns of differentassets. Our discussion emphasizes intuition and applications. For statistical theoryof multivariate time series analysis, readers are referred to Lütkepohl (1991) andReinsel (1993).8.1 WEAK STATIONARITY AND CROSS-CORRELATION MATRIXESConsider a k-dimensional time series r t = (r 1t ,...,r kt ) ′ .Theseriesr t is weaklystationary if its first and second moments are time-invariant. In particular, the meanvector and covariance matrix of a weakly stationary series are constant over time.Unless stated explicitly to the contrary, we assume that the return series of financialassets are weakly stationary.For a weakly stationary time series r t , we define its mean vector and covariancematrix asµ = E(r t ), Γ 0 = E[(r t − µ t )(r t − µ t ) ′ ], (8.1)where the expectation is taken element by element over the joint distribution of r t .The mean µ is a k-dimensional vector consisting of the unconditional expectationsof the components of r t . The covariance matrix Γ 0 is a k ×k matrix. The ith diagonalelement of Γ 0 is the variance of r it , whereas the (i, j)th element of Γ 0 is the covariancebetween r it and r jt . We write µ = (µ 1 ,...,µ k ) ′ and Γ 0 =[Ɣ ij (0)] when theelements are needed.8.1.1 Cross-Correlation MatrixesLet D be a k × k diagonal matrix consisting of the standard deviations of r it fori = 1,...,k. In other words, D = diag{ √ Ɣ 11 (0),..., √ Ɣ kk (0)}. The concurrent, orlag-zero, cross-correlation matrix of r t is defined asρ 0 ≡[ρ ij (0)] =D −1 Γ 0 D −1 .More specifically, the (i, j)th element of ρ 0 isρ ij (0) =Ɣ ij (0)√Ɣii (0)Ɣ jj (0) = Cov(r it, r jt )std(r it )std(r jt ) ,which is the correlation coefficient between r it and r jt . In a time series analysis,such a correlation coefficient is referred to as a concurrent, or contemporaneous,