2.1.2 Vector MA(oo) Re epresentation The first k rows of the vector system represented in(7)constitute a vector system yt+s +Et+s+业1Et+s-1+业2Et+8-2 +Fi1(y2-)+F12(y-1-p)+…+F1p(yt-p+1-p) ,=FI and Fl denotes the upper left block of Fi, where Fi is the matrix F raised to the jth power If the eigenvalues of f all lie inside the unit circle, then fs-oas s and y can be expressed as convergent sum of the history of E yt=+et+业1et-1+业2Et-2+业3et-3+…=+业(D)et The moving average matrices y, could equivalently be calculated as follows The operator更(L)(=Ik-重1L-重2L2 更LP)at(2)and业(Dat(9)are related by 业(L)=匝(L-1, requiring that Lk-1L一2L2一…-亞L[Lk+业1L+业2L2+…=I Setting the coefficient on L equal to the zero matrix produces Similarly, setting the coefficient on L equal to zero gives 业2=中1y1+更2 and in general for L 业。=重1亚-1+重2亚。-2+…+重2业。-p∫ors=1,2, (10) with业o= I k and业s=0fors<0 Note that the innovation in the MA(oo) representation in(9)is Et, the funda- mental innovation for yt. There are alternative moving average representations2.1.2 Vector MA(∞) Representation The first k rows of the vector system represented in (7) constitute a vector system: yt+s = µ + εt+s + Ψ1εt+s−1 + Ψ2εt+s−2 + ... + Ψs−1εt+1 +F s 11(yt − µ) + F s 12(yt−1 − µ) + .... + F s 1p (yt−p+1 − µ). Here Ψj = F (j) 11 and F (j) 11 denotes the upper left block of F j , where F j is the matrix F raised to the jth power. If the eigenvalues of F all lie inside the unit circle, then F s → 0 as s → ∞ and yt can be expressed as convergent sum of the history of ε: yt = µ + εt + Ψ1εt−1 + Ψ2εt−2 + Ψ3εt−3 + ... = µ + Ψ(L)εt . (9) The moving average matrices Ψj could equivalently be calculated as follows. The operator Φ(L)(= Ik − Φ1L − Φ2L 2 − ... − ΦpL p ) at (2) and Ψ(L) at (9) are related by Ψ(L) = [Φ(L)]−1 , requiring that [Ik − Φ1L − Φ2L 2 − ... − ΦpL p ][Ik + Ψ1L + Ψ2L 2 + ...] = Ik. Setting the coefficient on L 1 equal to the zero matrix produces Ψ1 − Φ1 = 0. Similarly, setting the coefficient on L 2 equal to zero gives Ψ2 = Φ1Ψ1 + Φ2, and in general for L s , Ψs = Φ1Ψs−1 + Φ2Ψs−2 + ... + ΦpΨs−p for s = 1, 2, ... (10) with Ψ0 = Ik and Ψs = 0 for s < 0. Note that the innovation in the MA(∞) representation in (9) is εt , the funda￾mental innovation for yt . There are alternative moving average representations 8