If the V AR(p) process is stationary, we can take expectation of both side of (1)to calculate the mean u of the process =c+重1+重2+…+重p, 更p) Equation(1)can then be written in terms of deviations from the mean (yt-p)=重1(yt-1-1)+重2y-2-1)+…+更p(y-p-1)+Et.(3) 2.1.1 Conditions for Stationarity As in the case of the univariate AR(p)process, it is helpful to rewrite(3)in terms of a VAR(1) process. Toward this end, define P×1) 重1重2更3 更 I00 O IK 0 F 0 0 000 000 0 00000 0
If the V AR(p) process is stationary, we can take expectation of both side of (1) to calculate the mean µ of the process: µ = c + Φ1µ + Φ2µ + ... + Φpµ, or µ = (Ik − Φ1 − Φ2 − ... − Φp) −1 c. Equation (1) can then be written in terms of deviations from the mean as (yt − µ) = Φ1(yt−1 − µ) + Φ2(yt−2 − µ) + ... + Φp(yt−p − µ) + εt . (3) 2.1.1 Conditions for Stationarity As in the case of the univariate AR(p) process, it is helpful to rewrite (3) in terms of a V AR(1) process. Toward this end, define ξt = yt − µ yt−1 − µ . . . yt−p+1 − µ (kp×1) (4) F = Φ1 Φ2 Φ3 . . . Φp−1 Φp IK 0 0 . . . 0 0 0 IK 0 . . . 0 0 0 0 0 . . . 0 0 0 0 0 . . . 0 0 0 0 0 . . . 0 0 0 0 0 . . . 0 0 (kp×kp) (5) and vt = εt 0 . . . 0 (kp×1) . 6
The V AR(p)in 3 )can then be rewritten as the following VAR(1) St=Fst which implies 5t+s=v++Fv+-1+F2v+s-2+…+F-v+1+F" where E(vv)9 foheruvise and 0 00 0 Q In order for the process to be covariance-stationary, the consequence of any given Et must eventually die out. If the eigenvalues of F all lie inside the unit circle, then the V aR turns out to be covariance-stationary Proposition The eigenvalues of the matrix F in(5) satisfy 2-2-…-重=0. Hence, a V AR(p)is covariance-stationary as long as a 1 for all values of A satisfying(8). Equivalently, the V AR is stationary if all values z satisfying 重2=0 lie outside the unit circle
The V AR(p) in (3) can then be rewritten as the following V AR(1): ξt = Fξt−1 + vt , (6) which implies ξt+s = vt+s + Fvt+s−1 + F 2vt+s−2 + ... + F s−1vt+1 + F s ξt , (7) where E(vtv 0 s ) = Q for t = s 0 otherwise and Q = Ω 0 . . . 0 0 0 . . . 0 . . . . . . . . . . . . . . . . . . 0 0 . . . 0 . In order for the process to be covariance-stationary, the consequence of any given εt must eventually die out. If the eigenvalues of F all lie inside the unit circle, then the V AR turns out to be covariance-stationary. Proposition: The eigenvalues of the matrix F in (5) satisfy Ikλ p − Φ1λ p−1 − Φ2λ p−2 − ... − Φp = 0. (8) Hence, a V AR(p) is covariance-stationary as long as |λ| < 1 for all values of λ satisfying (8). Equivalently, the V AR is stationary if all values z satisfying Ik − Φ1z − Φ2z 2 − ... − Φpz p = 0 lie outside the unit circle. 7
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 representations
2.1.2 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 fundamental innovation for yt . There are alternative moving average representations 8
based on vector white noise process other than Et. Let h denote a nonsingular (k x k) matrix, and define ut= hEt Then certainly ut is white noise E(ut) E(utu) HQ2H for t=T 0fort≠ Moreover, from( 9) we could writ H+H-Het+yiH-HEt-1+y2H-HEt-2+y3H-HEt-3+ u+Jout +J1ut-1+J2ut-2+J3ut-3+ where J。=yH1 (11) For example, H could be any matrix that diagonalizes n2 HSH=D with d a diagonal matrix. For such a choice of h, the element of ut are uncor- related with one another: a E(utu=HSH=D Thus, it is always possible to write a stationary V AR(p) process as a infinite moving average of a white noise vector ut whose elements are mutually uncorre- lated 2.1.3 Computation of Autocovariances of an Stationary V AR(p) Pro- cess We now consider to express the second moments for yt following a VAR(P) Recall that as in the univariate AR(p) process, the Yule-Walker equation are
based on vector white noise process other than εt . Let H denote a nonsingular (k × k) matrix, and define ut = Hεt . Then certainly ut is white noise: E(ut) = 0 and E(utu 0 τ ) = HΩH0 for t = τ 0 for t 6= τ . Moreover, from (9) we could write yt = µ + H−1Hεt + Ψ1H−1Hεt−1 + Ψ2H−1Hεt−2 + Ψ3H−1Hεt−3 + ... = µ + J0ut + J1ut−1 + J2ut−2 + J3ut−3 + ...., where Js = ΨsH−1 . (11) For example, H could be any matrix that diagonalizes Ω, HΩH0 = D, with D a diagonal matrix. For such a choice of H, the element of ut are uncorrelated with one another:a E(utu 0 t ) = HΩH0 = D. Thus, it is always possible to write a stationary V AR(p) process as a infinite moving average of a white noise vector ut whose elements are mutually uncorrelated. 2.1.3 Computation of Autocovariances of an Stationary V AR(p) Process We now consider to express the second moments for yt following a V AR(p). Recall that as in the univariate AR(p) process, the Yule-Walker equation are 9