obtained as 
o2 pe”) P 
s(w@) = —_————_ (5.209) 
2n Jace) P 
a2 LS Dy 3 yO eo 2 och fh BPO || ¢ 
= = (5.209’) 
| GO EGRET oe 3 atk GPO || @ 
5.3.4 Estimation of aj ... Qn, by . . . bn of ARMA(n,m) 
For starting values of a)- - - G,;, b,: - - b,, we can get a first approximation of the 
estimates from Eq. 5.208. Namely, with the use of n simultaneous equations on 
R(m+1), R(m+2)-+--R(m+n) fork =>m+1 
autoregressive parameters a; , d2,- - - a, can be estimated, although the less reliable val- 
ues of autocorrelation at larger m + n must be used. Then, inserting these values in the 
first (m + 1) equations in Eq. 5.208 on R(O), R(1) - - -R(m), m+ 1 unknown parameters 
by: - - bm and a2 can be estimated theoretically by solving the nonlinear equations on 
these parameters. 
The necessity for solving nonlinear equations can be shown as follows: In the 
expression for ARMA(n,m) 
Xp + QyXp1 t+ a2Xpot: + anXpn = €:t+ dD €-it- > -+dDm€im, 
using recursively 
€ py = Xp + OyXp2 + G2Xp3 t+ ++ An iXin t+ nXtr-1 
— by€,-2— b2€-3—- + > — DmEt-m-1 
X, = — X14 —42X,2° + + —ApXin 
+ bE + 07E 5-9 - - H Dpe em + er, 
X, = (—@1 + by)Xj-1 + — 2 —4b1)X-2 ++ + + + (— An — Gn_-101 Xin — And) Xi-r-1 
+1(b+ = bo )e 29 + (DD Dae nat bee ere (5.210) 
We still have the terms €,2,€,3- - : that should be expressed by X,2, X,3- - - and also 
€;-2,€,3. When the dependence of X; is expressed in terms of past X;, the equation will 
be nonlinear in the unknown parameters a), a2: - - and bj, bo, - - -, because their prod- 
ucts and squares are involved. Thus, the regression becomes nonlinear and requires a 
nonlinear least squares method for estimation. 
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