Akaike also made it clear that, for large N, FPE, and AIC are equivalent, because 
l+g/N 3 
q/ 62 
log FPE(q) = lo 
= log(1 + g/N) -log(1 —q/N) + log a? 
=logo2+2q/N (for large N). (5.258) 
Therefore AIC(q) = N log[FPE(q)]. 
When N, the number of observations, is large enough, the FPE criterion is the same as the 
AIC(q) criterion. 
BIC(q). Akaike***° also proposed the BIC(q) criterion, based on the Baysian con- 
cept, as a new criterion for determining the order. That is, 
BIC(q) = Nlog 62— (N —q)log(1—q/N)+qlogN 
aloe 
+ qlogsq tI ; (5.259) 
O¢ 
This equation can be modified as follows if we approximate 
((-N -q)log(1—q/N)} = {|-W-9)(-4/M)} =9-@°/N = 4 
for N >> q. Then 
€ 
a2 
BIC(q) = AIC(g) + gqlog N—1)+q oof 5 1 | } 
= [Nlog 2+ 2g] + qdogN)—q+qlogsg| 52-1] +b. (5.260) 
C, 
This relation shows that the difference between BIC(q) and AIC(q) is approximately 
one g of AIC(q) and is replaced with g log N. This replacement has the effect of increas- 
ing the weight attached to the penalty term which takes account of the number of 
parameters in the model. Shibata>® tells us that AIC(q) slightly overestimates and BIC(q) 
slightly underestimates the real q value. 
55.5 Examples of Order Determination through MAIC 
In Sections 5.2.1 to 5.2.5, examples of the synthesized processes AR(0), AR(1), 
AR(2), ARMA(2.1), MA(2), MA(1), and ARMA(2.2) generated by mathematical models 
were shown. When these processes were given as observed data, the orders N and M of 
each process were estimated by the MAIC method and then parameters a; - - - ay; 
b,- - - bm were estimated by the method described at each subsection. The results are 
198 
