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3.1 ARMA(p, q) Processes 87
{πj} given by (see (3.1.8))
π0 1,
π1 −(.4 + .5),
π2 −(.4 + .5)(−.4),
πj −(.4 + .5)(−.4) j−1 , j 1, 2,....
(A direct derivation of these formulas for {ψj} and {πj} was given in Section 2.3
without appealing to the recursions (3.1.7) and (3.1.8).)
Example 3.1.2 An AR(2) process
Let {Xt} be the AR(2) process
Xt .7Xt−1 − .1Xt−2 + Zt, {Zt} ∼WN 0,σ 2 .
The autoregressive polynomial for this process has the factorization φ(z) 1−.7z+
.1z 2 (1 − .5z)(1 − .2z), and is therefore zero at z 2 and z 5. Since these
zeros lie outside the unit circle, we conclude that {Xt} is a causal AR(2) process with
coefficients {ψj} given by
ψ0 1,
ψ1 .7,
ψ2 .7 2 − .1,
ψj .7ψj−1 − .1ψj−2, j 2, 3,....
While it is a simple matter to calculate ψj numerically for any j, it is possible also
to give an explicit solution of these difference equations using the theory of linear
difference equations (see TSTM, Section 3.6).
The option Model>Specify of the program ITSM allows the entry of any causal
ARMA(p, q) model with p