"Frontmatter". In: Analysis of Financial Time Series

VAR MODELS 3098.2 VECTOR AUTOREGRESSIVE MODELSA simple vector model useful in modeling asset returns is the vector autoregressive(VAR) model. A multivariate time series r t is a VAR process of order 1, or VAR(1)for short, if it follows the modelr t = φ 0 + Φr t−1 + a t , (8.8)where φ 0 is a k-dimensional vector, Φ is a k × k matrix, and {a t } is a sequence ofserially uncorrelated random vectors with mean zero and covariance matrix Σ. Inapplication, the covariance matrix Σ is required to be positive definite; otherwise,the dimension of r t can be reduced. In the literature, it is often assumed that a t ismultivariate normal.Consider the bivariate case [i.e., k = 2, r t = (r 1t , r 2t ) ′ and a t = (a 1t , a 2t ) ′ ]. TheVAR(1) model consists of the following two equations:r 1t = φ 10 + 11 r 1,t−1 + 12 r 2,t−1 + a 1tr 2t = φ 20 + 21 r 1,t−1 + 22 r 2,t−1 + a 2t ,where ij is the (i, j)th element of Φ and φ i0 is the ith element of φ 0 . Based onthe first equation, 12 denotes the linear dependence of r 1t on r 2,t−1 in the presenceof r 1,t−1 . Therefore, 12 is the conditional effect of r 2,t−1 on r 1t given r 1,t−1 .If 12 = 0, then r 1t does not depend on r 2,t−1 , and the model shows that r 1t onlydepends on its own past. Similarly, if 21 = 0, then the second equation shows thatr 2t does not depend on r 1,t−1 when r 2,t−1 is given.Consider the two equations jointly. If 12 = 0and 21 ̸= 0, then there is aunidirectional relationship from r 1t to r 2t .If 12 = 21 = 0, then r 1t and r 2t areuncoupled. If 12 ̸= 0and 21 ̸= 0, then there is a feedback relationship betweenthe two series.In general, the coefficient matrix Φ measures the dynamic dependence of r t .Theconcurrent relationship between r 1t and r 2t is shown by the off-diagonal element σ 12of the covariance matrix Σ of a t .Ifσ 12 = 0, then there is no concurrent linear relationshipbetween the two component series. In the econometric literature, the VAR(1)model in Eq. (8.8) is called a reduced-form model because it does not show explicitlythe concurrent dependence between the component series. If necessary, an explicitexpression involving the concurrent relationship can be deduced from the reducedformmodel by a simple linear transformation. Because Σ is positive definite, thereexists a lower triangular matrix L with unit diagonal elements and a diagonal matrixG such that Σ = LGL ′ ; see Appendix A on Cholesky Decomposition. Therefore,L −1 Σ(L ′ ) −1 = G.Define b t = (b 1t ,...,b kt ) ′ = L −1 a t .ThenE(b t ) = L −1 E(a t ) = 0, Cov(b t ) = L −1 Σ(L −1 ) ′ = L −1 Σ(L ′ ) −1 = G.