where O0=Ik. Thus any vector MA(q) process is covariance-stationary 1.4Ⅴ ector MA(∞)P rocess The vector MA(oo)process is written yt=l+Et+业1Et-1+业2Et-2+ where Et is a vector white noise process and y i denotes an(k x k) matrix of MA coefficients Definition: For an(n x m) matrix H, the sequence of matrices Hs so_ is absolutely summable if each of its elements forms an absolutely summable scalar sequence Example If i denotes the row i, column j element of the moving average parameters matrix ys associated with lag s, then the sequence ys=o is absolutely if ∑|e|<∞fori=1,2…, k and j=1,2,… Theorem. Let yt=+et+业1Et-1+业2Et-2+ where Et is a vector white noise process and (ilio is absolutely summable. Let lit denote the ith element of yt, and let u; denote the ith element of u. Then (a). the autocovariance between the ith variable at time t and the jth variable s period earlier, E(yit -Hi)(; t-s-ui), exist and is given by the row i, column j r ys+Qyy fa 0,1,2where Θ0 = Ik. Thus any vector MA(q) process is covariance-stationary. 1.4 Vector MA(∞) Process The vector MA(∞) process is written yt = µ + εt + Ψ1εt−1 + Ψ2εt−2 + .... where εt is a vector white noise process and Ψj denotes an (k ×k) matrix of MA coefficients. Definition: For an (n × m) matrix H, the sequence of matrices {Hs} ∞ s=−∞ is absolutely summable if each of its elements forms an absolutely summable scalar sequence. Example: If ψ (s) ij denotes the row i, column j element of the moving average parameters matrix Ψs associated with lag s, then the sequence {Ψs} ∞ s=0 is absolutely if X∞ s=0 |ψ (s) ij | < ∞ for i = 1, 2, ..., k and j = 1, 2, ..., k. Theorem: Let yt = µ + εt + Ψ1εt−1 + Ψ2εt−2 + .... where εt is a vector white noise process and {Ψl} ∞ l=0 is absolutely summable. Let yit denote the ith element of yt , and let µi denote the ith element of µ. Then (a). the autocovariance between the ith variable at time t and the jth variable s period earlier, E(yit − µi)(yj,t−s − µj), exist and is given by the row i, column j element of Γs = X∞ v=0 Ψs+vΩΨ0 v for s = 0, 1, 2, ...; 3