fxXD) =Z7 +bi\Z7" 1 + boZ™ 2 +- - -+b,,=0 (5.201) 
must have all its roots vj, v2- - - V» inside the unit circle or 
BQ =14+5;Z+boZ +- - -+b,,Z” 
must have all its roots outside the unit circle. 
5.3.2 Green’s Function for ARMA(n,m) 
Green’s function G; is defined as 
X,= DY Gj €j= Go €:+ Gi Ent: +Gj enjt- 
j=0 
= (Go+ GiB + GB? +--+ +G; Bi+- + -e;. 
Then Eq. 5.196’ is 
(1 +a,B +a2B?+- - -+a,B")(Go+ GiB + G2B? ++ - -+G; Bi+-- Je, 
=(1+b,B + boB? +--+ -+bm B”E;. (5.202) 
Equating the same order of backward shifting operator B for both sides of Eq. 5.202, 
Go=1 
a 1Go + G, = by 
arGo+ Gia, + G2 = b2 
QmGo ste G1Qm-1 + G2am-2 +---+G= bm. 
apGo + Giax_) + Goay2- - -+Gp=0 fork > m, (5.203) 
or 
Go = 1 
G, =b,-a, 
G2 = bz — (bi — a1)a; — a2 
Gin = bm, - + ¢—Ap. (5.204) 
Explicitly Green’s function can be obtained through the solution of the homogeneous 
equation 
X;+ aX1-1 + a2Xp2++ + an Xin =0 (5.205) 
with initial conditions given by Eq. 5.204, that is, in the form 
Gi= 21 M+ 92 WAt- + -4+8n Uh (5.206) 
where f;, 2° - -, are the eigenvalues or the roots of the characterization equation 
fi(Z) =Z" +.a;Z") + anZ** +- - - a, = 0. 
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