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 circleThe 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