e(0)=1 (5.163) 
(1—a?) (1—a,b, +b?) 
(a) (By) 
OD). —————— (5.164) 
(1—a,b}+b?) a(1—ap) 
(b1) (1 +a) 
o(r) =—ayo(r—1)—-ap9o(r-2) r=. (5.165) 
These are the autocovariance functions of ARMA(2.1). As was done for AR(2) in 
Section 5.2.3, we can get the same results as the solution of the homogeneous equation 
because Eq. 5.157 is the homogeneous equation with the same coefficient as the homoge- 
neous Eq. 5.126 or Eq. 5.60 of AR(2) expressed by Eq. 5.59. 
Its solution is in the shape of 
R(r) = By My t+ Bz by, (5.166) 
1,2 being the roots of characteristic equation 
fZ=Z2+a,Z+a,=0 (5.167) 
or 
Mi +2 =—-a 
Mi2 = 4. 
B, and Bare constants and are determined by the boundary conditions 
B, + Bz = R(O) 
Bi + Boy = R(1). 
After algebraic manipulation, 
&1 §2 
Bi = 081 Lan 
| Ga CSD) 
(5.167) 
82 81 
B= of @ + —_—————— 
ee 1nd) (l-m1 be) 
or, using the relations of Eqs. 5.135 and 5.136, 
+b +b +b 
Bile oe ew Gece en aan (5.168) 
(uy - me)? | (-“2) 1 -py wo 
142 
