1 
ize 5(-a =e J 4a) =—a,(« + Pai) 
We can transform Eq. 6.86 into the equation expressed by @,a@2, and b; of the corre- 
sponding ARMA(2.1) model by the relation of the so—called covariance equivalence. 
a. When at <4ay (k*<1): 
Further, for simplicity of expression, we set 
A, =-—utiv 
(6.88) 
An = —u-iv. 
Then 
CO eee el 
NA a 4a9- a] = >On Lore 
6.89 
1 (6.89) 
u=—Q\ 
Q,=2u 
(6.90) 
Qo = u? + v?. 
From the covariance equivalence relations between AR(2) and ARMA(2.1) we have 
Eqs. 6.77, 6.78, 6.79, and 6.80. Further use of Eq. 6.88 gives 
— Ay = fy + fz = EMAL 4g. HWIAT = oul. 2 cos(vAt) 
fy — M2 = eo“ 23 sin (vAt) 
1 —az = 1—pyn = eo“ -2 sinh (uA) 
L+ap=14+pyz =e’ -2 cosh (uAt) Cc) 
pi — a = 2e“' .i sin (2vAt) 
1 —a3 = 1 —ptud = 2e°*™ - sinh (2uAz). 
Inserting these expressions into Eq. 6.86 gives 
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