Actually, however, it is troublesome to solve the nonlinear equations, and other ap- 
proximation methods are used. An approximation method introduced by Pandit and Wu*° 
is to use the inverse function, taking advantage of the linearity between the unknown 
coefficient and the inverse function as shown in Eq. 5.150 for ARMA(2.1). In this meth- 
od, we first fit the pure autoregressive model of an appropriate large order AR(&) to this 
ARMA(n,m) process and then use the coefficients of this AR(k) model as an approxima- 
tion for inverse functions at larger values of k. Then from the equation that connects b; 
and J;, we can gradually improve the approximation of b; and a; and finally adjust the 
results from the viewpoint of invertibility. The procedure is rather complicated, and care 
is necessary to get good estimates. In any case estimation of the coefficients for ARMA 
(n,m) is not necessarily easy. 
Estimation of the parameters for the pure autoregressive process AR(n) is more 
straightforward, as we saw for AR(2) in Section 5.2.3 or as will be shown in Section 5.4, 
since it can be done through linear regression. 
In practical applications of the model fitting technique, the pure autoregressive pro- 
cess AR(k) is frequently used as the model to be fitted, although the order k sometimes 
becomes a little larger than n of ARMA(n,m), which should be the actual model to be 
fitted to the process under consideration. We saw this tendency in examples of 
ARMA(2.1), MA(2), MA(1), and ARMA(2. 2) shown in Figs. 5.27, 5.30, 5.34, 5.37, 
5.40, and 5.43. 
5.3.5 Adoption of ARMA(n, n— 1), ARMA(2n, 2n —1) 
As was mentioned in Sections 5.2.2 and 5.2.4, Green’s functions and the autocovari- 
ance functions are in the form for AR(1): Gj =a! =y'; R(r) = Bu! and for ARMA(2.1): 
Gj= 21 uw) + 22 wi; R(r) = Buj + Bz wu}, and show increasing complexity as the order 
increases. For AR(2), as derived in Section 5.2.3, G; and R(r) were of the same type as 
those for ARMA(2.1). However, as was mentioned in Section 5.2.4, ARMA(2.1) is rec- 
ommended by Pandit and Wu* as a more general process that is made up of two expo— 
nentials. The other reason for preferring ARMA(2.1) over AR(2) is the fact that the 
ARMA(2.1) process is more closely related to the system that is governed by the second 
order differential equation, as will be explained in Section 6.3.2. In the same way, for a 
more complex process made up of three exponentials, AR(3.2) is the most general pro- 
cess. Extending this relation, ARMA(n, n—1) is recommended by Pandit and Wu" as the 
most general process when the process is represented by nth order dynamics as 
Gj= er Miter Wht +n Meds RO) = By w+ Bo wht: - -Bybs. 
The reason that n — 1 should be the order of the moving average part is also shown in 
Pandit and Wu.*° Furthermore, as they pointed out, empirical experience indicates that it 
is better to increase n in steps of two and fit ARMA(2n, 2n — 1) models for n = 1, 2, 3. 
Also, if a complex root appears among #4;, and a), - - - are to be real, complex roots 
must appear as conjugate pairs as was shown for ARMA(2.1) in Section 5.2.4. Therefore 
to fitan ARMA(2n, 2n — 1) model is more practical than to fit an odd order model. 
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