5.4.5 Estimation of a2 for AR(n) 
N 
oe is estimated as the variance of the residual error €, from Q = yy. EP, 
t=n+l 
Because (N — n) observations are summed and n + 1 parameters (u and na;’s) have been 
estimated, the unbiased estimation is 
1 
a2 A A A A 
— Omaha a 
€ (N—n)—(n a eae 1,42 D) 
: S {x-¥) X.4-X) Jina 
= = + _j- tee et mA ye 
omni al Q\\A;-] AylA; 
=" _[r@) +4 RG ee R(n)| (5.246) 
7 Neanest : ; 
When n <<N, 
G2 = R(0) + a,R(1) + + -+ anR(n) (5.246’) 
This result is the same as we obtained from the first equation of Eq. 5.220. 
5.5 DETERMINATION OF THE ORDER OF THE FITTED MODEL 
As was explained in the preceding sections, the process becomes more complicated 
as the order increases. Thus AR(2) is more complicated than AR(1) and ARMA(3.2) is 
more complicated than ARMA(2.1). Estimation of parameters has been discussed in 
each section, so here only the determination of the order will be discussed. Several 
different methods have been described by Priestley.2*? The MAIC method developed by 
H. Akaike is now considered to be the most reasonable from the point of view of statisti- 
cal considerations and of information theory and is explained and recommended in the 
following section. 
5.5.1 Residual Error Method 
The change of residual error, expressed for example by Eq. 5.246’ for AR(n), is 
investigated. If the order is far smaller than the true order, o2, the residual error, will 
decrease considerably as the order increases from j ton. From there on, the decrease in 
residual error may not be significant but will remain at the same level. Point n is adopted 
as the proper order of the model. 
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