Springer - Read

Figure D-12
The PACF of the model in
Example D.3.5 together
with the sample PACF
of the transformed
AIRPASS.TSM series.
D.6 Model Properties 419
PACF
-0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0
0 10 20
Lag
30 40
D.6.3 Model Representations
As indicated in Section 3.1, if {Xt} is a causal ARMA process, then it has an MA(∞)
representation
∞
Xt ψjZt−j, t 0, ±1, ±2,...,
j0
where ∞
|ψj| < ∞ and ψ0 1.
j0
Similarly, if {Xt} is an invertible ARMA process, then it has an AR(∞) representation
∞
Zt πjXt−j, t 0, ±1, ±2,...,
j0
where ∞
j0 |πj| < ∞ and π0 1.
For any specified causal ARMA model you can determine the coefficients in these
representations by selecting the option Model>AR/MA Infinity. (If the model is not
invertible, you will see only the MA(∞) coefficients, since the AR(∞) representation
does not exist in this case.)
Example D.6.3 The current subset MA(23) model for the transformed series AIRPASS.TSM does
not have an AR(∞) representation, since it is not invertible. However, we can replace