5.2.5 ARMA(I.1), MA(2) ARMA(I 2) MA(1), and ARMA(2.2) 
525.1 ARMA(I.1). In ARMA(2.1), expressed by Eq. 5.120 
X; + a, Xj) + 02 X-2 =€;+ DE 1-1, 
when a2 = 0, the process is called ARMA(1.1). We can easily get Green’s function, the 
inverse function, the autocovariance, and the spectrum function of ARMA(1.1) by modi- 
fying those functions for ARMA(2.1) by setting a2 = 0. 
525.2 MA(2). When a; =a2=0 in Eq. 5.120 and there exist b2 in addition to bj, 
X,=€;+ by €,1+b2 €p2. (5.184) 
This process is called MA(2). 
In this case Green’s function is the function for expressing X; by MA (~), so to 
speak of Green’s function for a pure moving average process has no meaning. However, 
formally, 
xX = » G; €,_j 
j=0 
and 
Go=1 
G,=b; 
G2 = b2 
Gj=0 j23. 
In this case the inverse function for expressing X, by AR () is meaningful. From Eq. 
5.184, 
X, = (1+ DB + bpB*) €, 
= B(B)E,, (5.184’) 
here 
BZ = 1+ di\Z + b2Z?. (5.185) 
Now supposing the quadrant 
fo(Z) = Z* +b\Z+b2 =0 (5.186) 
has two roots v; andv2, f2(Z) = (Z—v1)(Z—¥v2) 
1 ae 
V1,V2 =5(-01 + Fi 4b2| 6 (5.187) 
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