5.4 GENERAL AUTOREGRESSIVE PROCESS AR(n) 
5.4.1 Adoption of AR(n) Model 
The general autoregressive model 
X,+@)X,1 + a2X2+-- -+a,=€; (5.211) 
is considered to be a special case of the general autoregressive moving average model 
ARMA(n,m) when m = 0. If invertibility is satisfied for the ARMA(n,m) model, 
€,=| > -1B IX, 
j=0 
SO Ii AGI 2G ee 9 ole Gea 8 a, 
This expression means that ARMA(n,m) is transformed into an autoregressive model of 
infinite order AR( ). Practically, if a large enough n is taken, almost all of the ARMA 
(n,m) process can be approximated by an AR(n’) process if we assume invertibility. 
The general character of the AR(n) process will be summarized here, as AR(n) is 
the most common model used in the practical application of model fitting techniques as 
was already mentioned in Section 5.3.4. The estimation of parameters is much easier for 
the AR(n) process than for the MA(m) or AR(n,m) processes. 
5.4.2 Green's Function of AR(n) 
The AR(n) process is expressed as 
X,+ 4X1 +Q2X,2+-- -+a,=€; (5.211) 
(1+a)B +a7B7+- - -+a,B")X, = €, (5.211) 
a(B)X, = €;. (5.211”) 
Here 
Q@(Z) =1+a;Z+a,Z*+- - -a, Z". (5.212) 
Using Green’s function G; gives 
ive} 
X,= ». G; €-j= SG Bi €,=Got+Gy €11+ Gr€p2+: - -+G; €rj 
j=0 
jJ=0 
= (G,+ GiB +G2B* +--+ -+GBi+-- Je, (5.213) 
ee (S282) 
(1 +a,B + a2B trois -+ 7B”) 
Equating the same power of B on both sides, 
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