"Frontmatter". In: Analysis of Financial Time Series

318 VECTOR TIME SERIESis e h (1) = a h+1 . The covariance matrix of the forecast error is Σ. Ifr t is weaklystationary, then the l-step ahead forecast r h (l) converges to its mean vector µ as theforecast horizon l increases.In summary, building a VAR model involves three steps: (a) use the test statisticM(i) or the Akaike information criterion to identify the order, (b) estimate the specifiedmodel by using the least squares method (in some cases, one can reestimatethe model by removing statistically insignificant parameters), and (c) use the Q k (m)statistic of the residuals to check the adequacy of a fitted model. Other characteristicsof the residual series, such as conditional heteroscedasticity and outliers, can also bechecked. If the fitted model is adequate, then it can be used to obtain forecasts.8.3 VECTOR MOVING-AVERAGE MODELSA vector moving-average model of order q, orVMA(q), is in the formr t = θ 0 + a t − Θ 1 a t−1 −···−Θ q a t−q or r t = θ 0 + Θ(B)a t , (8.21)where θ 0 is a k-dimensional vector, Θ i are k × k matrixes, and Θ(B) = I − Θ 1 B −··· − Θ q B q is the MA matrix polynomial in the back-shift operator B. Similarto the univariate case, VMA(q) processes are weakly stationary provided that thecovariance matrix Σ of a t exists. Taking expectation of Eq. (8.21), we obtain thatµ = E(r t ) = θ 0 . Thus, the constant vector θ 0 is the mean vector of r t for a VMAmodel.Let ˜r t = r t − θ 0 be the mean-corrected VAR(q) process. Then using Eq. (8.21)and the fact that {a t } has no serial correlations, we have1. Cov(r t , a t ) = Σ,2. Γ 0 = Σ + Θ 1 ΣΘ ′ 1 +···+Θ qΣΘ q ′ ,3. Γ l = 0 if l>q, and4. Γ l = ∑ qj=l Θ jΣΘ ′ j−lif 1 ≤ l ≤ q, whereΘ 0 =−I.Since Γ l = 0 for l>q, the cross-correlation matrixes (CCM) of a VMA(q) processr t satisfyρ l = 0, l > q. (8.22)Therefore, similar to the univariate case, the sample CCMs can be used to identifythe order of a VMA process.To better understand the VMA processes, let us consider the bivariate MA(1)modelr t = θ 0 + a t − Θa t−1 = µ + a t − Θa t−1 , (8.23)where, for simplicity, the subscript of Θ 1 is removed. This model can be writtenexplicitly as