Especially when m = n—1, namely for ARMA(n, n — 1), by Pandit and Wu*° 
(uz 1+ byw + - - - +bp-1) 
BO eee 
" (Wi—M)Ui— M2) + + Ui — Me) Ui—Mis dd «+ Hi-Ln) 
US Mor 2 81g (5.207) 
where the denominator is the product of all terms (u;—;) for j= 1,2, - - - n, excluding 
the zero term (u;—;). Equations 5.135 and 5.136 are the special cases j = 1 and 2 for 
n=2. Each real root yu; in Eq. 5.206 provides an exponentally dynamic mode like an 
AR(1) model, whereas a complex conjugate pair of roots as in Eq. 5.137’ gives an expo- 
nentially decaying sinusoidal mode, whose frequency and decay rate can be obtained as 
Egs. 5.140 and 5.139 illustrated in Fig. 5.24 in Section 5.2.4. 
5.3.3 Autocovariance and Spectrum Function of ARMA(n,m) 
For the ARMA(n,m) process, 
X, = — 4X1 —A2Xp-2—+ + ApXtnt+€r+ dy €-1+ 52 €p-2+- > -+dm Erm. 
Here, taking into account the relations 
X,= YG €.j=Go €:+ Gy €-1+ Gz rat: -+GjEpjt+- -- 
Xp = SG Sp pap SO Gorn Go ara Gy see 5 oe (Ge Gra nee % 6 
Ele,- €,-+] = 0, oe ’ 
multiplying both sides of these upper two equations term by term, respectively, and tak- 
ing the expected values gives, 
when r=0, 
R(O) = —a,R(1)—a2(R)(2) —- - -—a,R(n) + (G + biG; + b2G2 +: - - bmGm)oe 
when r= 1, 
R(1) = —a@;R(0) — a2R(1) — a3R(2) —- + -—a,R(n—-1) 
+ (b}Go + boG + b3G2+- > - + DmGm-1)02 
RQ)=-----:: 
when r =m, 
Rim) = —a,R(m—1)—a2R(m— 2) +: - -a, R(m—-n) 
+Dm Go a2 
and 
R(k) =—aiR(k-—1)—aoR(k—2)+- - -+a,R(k—n). k => m+1. (5.208) 
Analogous to the case of ARMA(2.1), R(0) to R(k),k = n+1, can be obtained by solving 
these simultaneous equations. Also, if necessary, we can get the general expression for 
R(r),k = m+1, by generalizing the case of ARMA(2.1). By the same procedure we 
used for ARMA(2.1), the spectrum s(w) for the ARMA(n,m) process is naturally 
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