€, is now a purely uncorrelated random process, 
Bles-eul=| iit (5.119) 
From Eqs. 5.117 and 5.118 
X,— a1 X11 — 43 X-2 = DE-1 + €; 
this equation is expressed generally, changing — a,’ toa, and—ay’ toa, as 
X, + A,X; + A2X1_-2 = Dj € 1 + €;. (5.120) 
When the process X; is expressed by Eq. 5.120, where aj, a2, and b, are constants and 
€,1S arandom variable, this process X, is called a second order autoregressive first order 
moving average process, ARMA(2.1). 
Equation 5.120 can be written as 
€,= X;+ a)X;_) + €2X;9 — dD €;-} (5.120”) 
This expression shows that, in order to compute the present value of€,, we neede€,_ , and 
when we compute €, recursively, we need the preceding values of €,_; starting from the 
beginning €;2, €;3- - -. This situation is different from AR(1) and AR(2) as shown by 
Eq. 5.7 and Eq. 5.59 for which we do not need any preceding values of €,.;. This makes 
the estimation of the ARMA(2.1) model much more difficult than that of the AR(1) and 
AR(2) models. This difficulty can also be shown as follows. 
From Eq. 5.120’ 
€y-1 = Xj + a1 Xp2 + 2X,_3 — Dy E_2. (5.121) 
Inserting this into Eq. 5.120 gives 
X= — ,Xj_-1 — A2Xj_2 + bilXi1 + a;X;_2 + a2X 1-3 — Di€ 2} Gi 
X= (— a + by)X1 + (a2 + ayb1) Xo + a2b1X,3-Dier2+€r- (5.122) 
Here €;_ 1s still included, so it should be expressed in the same way as Eq. 5.121 by X;_2, 
X-3: - + ande,3, and soon. However, even with Eq. 5.122, when the dependence of X, 
is expressed in terms of past X,, the equation is nonlinear in the unknown parameters a), 
a2, and by. As a result, the regression becomes nonlinear and requires a nonlinear least 
squares method for estimation, which is different from the case of AR(1), AR(2) or the 
general AR model as will be shown in Section 5.4, where all coefficients can be obtained 
by the linear least squares method. The estimation of a), a2, b; for ARMA(2.1) will be 
shown later. 
5.2.4.1 Green’s Function for ARMA(2.1). Equation 5.120 is expressed, using the 
backward shift operator B, as 
(1 + a,B + aB®)X, = (1+ bi B)e;. (5.123) 
135 
