and taking transpose r=E[(y2-)34+-p) 1) 1.2 Vector white noise process Definition a k x I vector process Et, tET) is said to be a white-noise process if (). E(et) (ii). E(EET) 0ift≠ where S2 is an(k x k) symmetric positive definite matrix. It is important to note that in general n2 is not necessary a diagonal matrix, since it is the contempora- neous correlation among variables that called for the needs of vector time series anaIvsis 1.3 Vector MA(q)Process A vector moving average process of order g takes the form yt=p+et+1et-1+白2et-2+…+eet-q, where Et is a vector white noise process and e; denotes an(k x k)matrix of MA coefficients for j=1, 2,.. 9. The mean of yt is u, and the variance is ro=E[(yt-1)(y:-) fEE/+O1EEt-1Et10+E2EEt-2Et-2Je? +…+nEet-9=-le 2+192e1+e2962+…+22e with autocovariance(compares with )j of Ch. 14 on p. 3) e!+6+19261++2!2e2+…+ese for j=1, 2, r={se-+19+1+2+2+…++9e4foj for liland taking transpose, Γ 0 j = E[(yt − µ)(yt+j − µ) 0 ] = E[(yt − µ)(yt−(−j) − µ) 0 ] = Γ−j . 1.2 Vector White Noise Process Definition: A k × 1 vector process {εt , t ∈ T } is said to be a white-noise process if (i). E(εt) = 0; (ii). E(εtε 0 τ ) =  Ω if t = τ 0 if t 6= τ, where Ω is an (k × k) symmetric positive definite matrix. It is important to note that in general Ω is not necessary a diagonal matrix, since it is the contempora￾neous correlation among variables that called for the needs of vector time series analysis. 1.3 Vector MA(q) Process A vector moving average process of order q takes the form yt = µ + εt + Θ1εt−1 + Θ2εt−2 + ... + Θqεt−q, where εt is a vector white noise process and Θj denotes an (k ×k) matrix of MA coefficients for j = 1, 2, ..., q. The mean of yt is µ, and the variance is Γ0 = E[(yt − µ)(yt − µ) 0 ] = E[εtε 0 t ] + Θ1E[εt−1ε 0 t−1 ]Θ0 1 + Θ2E[εt−2ε 0 t−2 ]Θ0 2 +... + ΘqE[εt−qε 0 t−q ]Θ0 q = Ω + Θ1ΩΘ0 1 + Θ2ΩΘ0 2 + ... + ΘqΩΘ0 q , with autocovariance (compares with γj of Ch. 14 on p.3) Γj =    ΘjΩ + Θj+1ΩΘ0 1 + Θj+2ΩΘ0 2 + ... + ΘqΩΘ0 q−j for j = 1, 2, ..., q ΩΘ0 −j + Θ1ΩΘ0 −j+1 + Θ2ΩΘ0 −j+2 + ... + Θq+jΩΘ0 q for j = −1, −2, ..., −q 0 for |j| > q, 2