Relation of A(2) to ARMA(2.1) by Covariance Equivalence. As was discussed in 
Section 5.3.5, ARMA(2.1) is more general and more flexible than AR(2). The Green’s 
function for AR(2) is composed of two exponentials as G; = emul + golly that is one step 
extension or complication of AR(1) process where the Green’s function is G; = pw that is 
one exponential. By the same token, we can say that ARMA(2.1) corresponds more 
closely or more generally and uniquely with A(2) than AR(2) does with A(2). 
Now let us express the ARMA(2.1) that corresponds with A(2) as Eq. 5.120 
X1+ aX; + A2X 1-2 = €, + DE 1-1, 
or as Eq. 5.123, 
(1+ a,B + a2B*)X, = (1 + bi B)e;. 
The autocovariance function for this ARMA(2.1) is from Eq. 5.166, 
R(r) = By + Boys, 
where /4;,//2 are the roots of the characteristic equation equated to zero for ARMA(2.1). 
Here 
fZ =Z? +a;Z+a2=0. 
From Eqs. 5.67 and 5.75 
My +2 =—-ay 
Mif2 = 42, 
(-«: =e ai —4a:) j 
Nile 
41,2 = 
and from Eqs. 5.168 and 5.169 
p, = 2 thy) | tbr woth 
(ui—M2)? | 1-wt 1 -wae 
_ Fein tbr) | wotbi mith 
(ui—M2)” | 1-ws 1—-pyr 
On the other hand, for A(2), when s = rAt, from Eq. 6.71, 
2 
OZ A rAt A rAt 
R At - 1 — 2 
ey) 2 A2(A4 73) G2 A ) 
220 
