Springer - Read

266 Chapter 8 State-Space Models
To express the local linear trend model in state-space form we introduce the state
vector
Xt (Mt,Bt) ′ .
Then (8.2.4) and (8.2.5) can be written in the equivalent form
Xt+1
1
0
1
1
Xt + Vt, t 1, 2,..., (8.2.6)
where Vt (Vt,Ut) ′ . The process {Yt} is then determined by the observation equation
Yt [1 0] Xt + Wt. (8.2.7)
If {X1,U1,V1,W1,U2,V2,W2,...} is an uncorrelated sequence, then equations (8.2.6)
and (8.2.7) constitute a state-space representation of the process {Yt}, which is a
model for data with randomly varying trend and added noise. For this model we have
v 2, w 1,
F
1
0
1
1
, G [1 0], Q
Example 8.2.2 A seasonal series with noise
σ 2
v
0
0 σ 2
u
, and R σ 2
w .
The classical decomposition (1.5.11) expressed the time series {Xt} as a sum of
trend, seasonal, and noise components. The seasonal component (with period d)was
a sequence {st} with the properties st+d st and d t1 st 0. Such a sequence can
be generated, for any values of s1,s0,...,s−d+3, by means of the recursions
st+1 −st −···−st−d+2, t 1, 2,.... (8.2.8)
A somewhat more general seasonal component {Yt}, allowing for random deviations
from strict periodicity, is obtained by adding a term St to the right side of (8.2.8),
where {Vt} is white noise with mean zero. This leads to the recursion relations
Yt+1 −Yt −···−Yt−d+2 + St, t 1, 2,.... (8.2.9)
To find a state-space representation for {Yt} we introduce the (d − 1)-dimensional
state vector
Xt (Yt,Yt−1,...,Yt−d+2) ′ .
The series {Yt} is then given by the observation equation
Yt [1 0 0 ··· 0] Xt, t 1, 2,..., (8.2.10)
where {Xt} satisfies the state equation
Xt+1 F Xt + Vt, t 1, 2 ..., (8.2.11)