obtained by postmultiplying (3) with(yt-i-u)and taking expectations. For 0. using t= t To= E(y-u(y-u 重1E(y-1-p)(yt-p)2+重2E(y=2-)(yt-1) +…+重,E(y1-p-1)(yt-1)+Eeyt 更1-1+重2r-2+…+重nr-p+ 更11+更2r+…+重r+9 and for j>0 r=中1Ty-1+2I-2+…+pTj-p These equations may be used to compute the t; recursively for j≥pif更1…重p and t To are known Let st be as defined in(4)and let 2 denote the variance of s, E(E!) yt-1- E To TI TI Postmultiplying(4) by its own transpose and taking expectation gives E划=E(F-1+v)(F-1+v)=FE(E1-1-1)F+E(vv) ∑=F∑F+Qobtained by postmultiplying (3) with (yt−j − µ) 0 and taking expectations. For j = 0, using Γj = Γ 0 −j , Γ0 = E(yt − µ)(yt − µ) 0 = Φ1E(yt−1 − µ)(yt − µ) 0 + Φ2E(yt−2 − µ)(yt − µ) 0 +... + ΦpE(yt−p − µ)(yt − µ) 0 + Eεt(yt − µ) 0 = Φ1Γ−1 + Φ2Γ−2 + ... + ΦpΓ−p + Ω = Φ1Γ 0 1 + Φ2Γ 0 2 + ... + ΦpΓ 0 p + Ω and for j > 0, Γj = Φ1Γj−1 + Φ2Γj−2 + ... + ΦpΓj−p. (12) These equations may be used to compute the Γj recursively for j ≥ p if Φ1,...,Φp and Γp−1,...,Γ0 are known. Let ξt be as defined in (4) and let Σ denote the variance of ξt , Σ = E(ξtξ 0 t ) = E            yt − µ yt−1 − µ . . . yt−p+1 − µ         × (yt − µ) 0 (yt−1 − µ) 0 . . . (yt−p+1 − µ) 0 0    =         Γ0 Γ1 . . . Γp−1 Γ 0 1 Γ0 . . . Γp−2 . . . Γ 0 p−1 Γ 0 p−2 . . . Γ0         . Postmultiplying (4) by its own transpose and taking expectation gives E[ξtξ 0 t ] = E[(Fξt−1 + vt)(Fξt−1 + vt) 0 ] = FE(ξt−1ξ 0 t−1 )F 0 + E(vtv 0 t ) or Σ = FΣF0 + Q. (13) 10