5.2.5.4 ARMA(12) Process. The equation for the ARMA(1.2) process is 
X,+a1X)_1 = E;+ by €,1+b2 € +2; (5.195) 
or, using the backward shift operator, 
(1 + a;B)X, = (1+ b1B + b2B7)€;. (5.195’) 
Analogous to ARMA(2.1), the stationality is determined by the left-hand side of this 
equation, i.e., by the autoregression part. On the other hand, the invertibility is deter- 
mined by the character of the right-hand side of Eq. 5.195’. Accordingly, the stationality 
is the same as for AR(1) and the invertibility is the same as for MA(2). Green’s function, 
the inverse function, the autocovariance function, and the spectrum function can be ob- 
tained in the same way as for ARMA(2.1). 
5.3. GENERAL ARMA(n,.m) PROCESS 
53.1 General ARMA(n,m) Process, Its Stationality and Invertibility 
Analogous to the preceding section, when 
X, + ayXe1 + QnXp2t- + OXpn = Ep ty E4452 Exnt- > -+dmErm, 
(5.196) 
the process X, is referred to as the mth order autoregressive, mth order moving average 
process, ARMA(n,m). When the backward shift operator B is used, 
(1+a,B+a,B2+- --+a, B)X,=(1+b,B + by B?+- - -+b,B™)€, (5.196’) 
a(B)X, = BiB, ; (5.196”’) 
where 
Q(Z) =1+a;Z+aZ* +- - -+a,Z" 
BZ) = 14 b1Z + eZ? +--+ BZ”. Cra 
Green’s function is derived from 
= BB) 
X,= 2 Gj Enj= GB)! 3 (5.198) 
j=0 
and the inverse function is derived as 
= a(B) 
€;= —1;, X_;=——-X,, (5.199) 
t 2, j4cj BB) t 
By the same logic as we had for ARMA(2.1), for this process to be stable, the char- 
acteristic function 
f(Z) = Z" +a,Z™ 1 + agZ** ++ + ++. ay (5.200) 
equated to 0, i.e., f;(Z) = 0 must have all its roots #;,42,- - -, inside the unit circle, 
ora(Z)=1+a\Z+ aoZ* +--+-++a,=0 must have all its roots outside the unit circle. 
Conversely, for this process to be invertible, the characteristic function 
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