If we express €, by an inverse function as 
€,= > -Ij Xj = (Clo-B- IB? ---- 1; B+), 
j=0 
then Eq. 5.123 becomes (1 + a)B + a2B”)X, = (1 +b)B)(—1,B — nB? —- -- I; B!- + -)X,. 
Comparing the coefficients of the powers of B gives 
“Jai (5.150) 
a,=b,-l, 
G7 Dilla 
ee (5.151) 
0=—-J;—by;1,j23 
or 
I, = b,-a, 
Iz = —a2—b1(b; —a)) (5.151’) 
Tj = — byl;-1, j=3- 
The last equation of Eq. 5.151’ is (1 + b; B) J; =O for j = 3. 
Therefore T;=—b, Ij. (5.152) 
This Eq. 5.152 is the recursive equation for Jj. 
5.2.4.3 Stationality and Invertibility of ARMA(2.1). The stationality of ARMA(2.1) 
depends on the convergence of the free oscillation that comes from the complementary 
function or the solution of the homogeneous equation. The homogeneous equation of 
ARMA(2.1) is the same as the homogeneous equation of AR(2), Eq. 5.60, which is 
X; + a1Xi-1 + a2X 1-2 = 0. 
Accordingly, the relations required of the coefficients a, and az for the ARMA(2.1) pro- 
cess to be stationary are the same as for AR(2), as has already been shown in Section 
5.2.3 and in Figs. 5.15 and 5.16. 
On the other hand, the invertibility of ARMA(2.1) depends on the boundedness 
of €;,€; = Sy -lj E,_;> when j — © and from the relation derived as Eq. 5.152, 
j=0 
J; = — b; J;-, \bl should be less than 1. This requirement corresponds to the character 
required for a, of AR(1), la;| < 1, so that the process AR(1) is stationary when 
Gj = a1 G-1. 
5.2.4.4 Autocovariance Function for ARMA(2.1). From Eq. 5.120 
X, = — A1X;-1 — A2X,-2 + €,+ Di E11: 
Therefore, multiplying both sides by X,_, and taking the expected values, 
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