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102 Chapter 3 ARMA Models
and
rn → 1.
Algebraic calculation of the coefficients θnj and rn is not feasible except for very simple
models, such as those considered in the following examples. However, numerical
implementation of the recursions is quite straightforward and is used to compute
predictors in the program ITSM.
Example 3.3.1 Prediction of an AR(p) process
Applying (3.3.7) to the ARMA(p, 1) process with θ1 0, we easily find that
ˆXn+1 φ1Xn +···+φpXn+1−p, n ≥ p.
Example 3.3.2 Prediction of an MA(q) process
Applying (3.3.7) to the ARMA(1,q) process with φ1 0gives
ˆXn+1
min(n,q)
j1
θnj
Xn+1−j − ˆXn+1−j
, n ≥ 1,
where the coefficients θnj are found by applying the innovations algorithm to the covariances
κ(i,j) defined in (3.3.3). Since in this case the processes {Xt} and {σ −1 Wt}
are identical, these covariances are simply
κ(i,j) σ −2 γX(i − j)
q−|i−j|
Example 3.3.3 Prediction of an ARMA(1,1) process
If
r0
θrθr+|i−j|.
Xt − φXt−1 Zt + θZt−1, {Zt} ∼WN 0,σ 2 ,
and |φ| < 1, then equations (3.3.7) reduce to the single equation
ˆXn+1 φXn + θn1(Xn − ˆXn), n ≥ 1.
To compute θn1 we first use Example 3.2.1 to find that γX(0)σ 2 1 + 2θφ + θ 2 / 1−
φ2 . Substituting in (3.3.3) then gives, for i, j ≥ 1,
⎧ 2
1 + 2θφ + θ / 1 − φ 2 , i j 1,
⎪⎨ 1 + θ
κ(i,j)
⎪⎩
2 , i j ≥ 2,
θ, |i − j| 1,i ≥ 1,
0, otherwise.