Springer - Read

100 Chapter 3 ARMA Models
3.3 Forecasting ARMA Processes
The innovations algorithm (see Section 2.5.2) provided us with a recursive method
for forecasting second-order zero-mean processes that are not necessarily stationary.
For the causal ARMA process
φ(B)Xt θ(B)Zt, {Zt} ∼WN 0,σ 2 ,
it is possible to simplify the application of the algorithm drastically. The idea is to
apply it not to the process {Xt} itself, but to the transformed process (cf. Ansley, 1979)
Wt σ −1 Xt, t 1,...,m,
Wt σ −1 (3.3.1)
φ(B)Xt, t > m,
where
m max(p, q). (3.3.2)
For notational convenience we define θ0 : 1 and θj : 0 for j>q. We shall also
assume that p ≥ 1 and q ≥ 1. (There is no loss of generality in these assumptions,
since in the analysis that follows we may take any of the coefficients φi and θi to be
zero.)
The autocovariance function γX(·) of {Xt} can easily be computed using any of
the methods described in Section 3.2.1. The autocovariances κ(i,j) E(WiWj),
i, j ≥ 1, are then found from
⎧
σ
⎪⎨
κ(i,j)
−2 γX(i − j), 1 ≤ i, j ≤ m
σ −2
p
γX(i − j)− φrγX(r −|i − j|) , min(i, j) ≤ mm,
r0 ⎪⎩
0, otherwise.
Applying the innovations algorithm to the process {Wt} we obtain
⎧
n
⎪⎨
⎪⎩
ˆWn+1
ˆWn+1
j1
θnj (Wn+1−j − ˆWn+1−j), 1 ≤ n