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268 Chapter 8 State-Space Models
where {Zt} ∼WN 0,σ2 , and φ(z) : 1 − φ1z −···−φpz p is nonzero for |z| ≤1.
To express {Yt} in state-space form we simply introduce the state vectors
⎡ ⎤
Yt−p+1
⎢ ⎥
⎢ Yt−p+2 ⎥
Xt ⎢ ⎥
⎢ ⎥,
t 0, ±1,.... (8.3.2)
⎣ . ⎦
Yt
From (8.3.1) and (8.3.2) the observation equation is
Yt [0 0 0 ··· 1]Xt, t 0, ±1,..., (8.3.3)
while the state equation is given by
⎡
⎤ ⎡ ⎤
0
⎢ 0
⎢
Xt+1 ⎢ .
⎢
⎣ 0
1
0
.
0
0
1
.
0
···
···
. ..
···
0 0
⎥ ⎢ ⎥
0 ⎥ ⎢ 0.
⎥
⎥ ⎢ ⎥
.
⎥Xt
⎥ + ⎢ ⎥
⎢ ⎥Zt+1,
1
⎥ ⎢
⎦ ⎣ 0
⎥
⎦
1
t 0, ±1,.... (8.3.4)
φp φp−1 φp−2 ··· φ1
These equations have the required forms (8.1.10) and (8.1.11) with Wt 0 and
Vt (0, 0,...,Zt+1) ′ , t 0, ±1,....
Remark 1. In Example 8.3.1 the causality condition φ(z) 0 for |z| ≤1 is equivalent
to the condition that the state equation (8.3.4) is stable, since the eigenvalues
of the coefficient matrix in (8.3.4) are simply the reciprocals of the zeros of φ(z)
(Problem 8.3).
Remark 2. If equations (8.3.3) and (8.3.4) are postulated to hold only for t
1, 2,...,and if X1 is a random vector such that {X1,Z1,Z2,...} is an uncorrelated
sequence, then we have a state-space representation for {Yt} of the type defined
earlier by (8.1.1) and (8.1.2). The resulting process {Yt} is well-defined, regardless
of whether or not the state equation is stable, but it will not in general be stationary.
It will be stationary if the state equation is stable and if X1 is defined by (8.3.2) with
Yt ∞
j0 ψjZt−j, t 1, 0,...,2 − p, and ψ(z) 1/φ(z), |z| ≤1.
Example 8.3.2 State-space form of a causal ARMA(p, q) process
State-space representations are not unique. Here we shall give one of the (infinitely
many) possible representations of a causal ARMA(p,q) process that can easily be
derived from Example 8.3.1. Consider the ARMA(p,q) process defined by
φ(B)Yt θ(B)Zt, t 0, ±1,..., (8.3.5)
where {Zt} ∼WN 0,σ2 and φ(z) 0 for |z| ≤1. Let
r max(p, q + 1), φj 0 for j>p, θj 0 for j>q, and θ0 1.