426 Structural Time Series Models
and the corresponding detrending filters are:
ψ ̃t|∞= λ|^1 −L|41 +λ| 1 −L|^4yt, ψ ̃t|t=θ(L)−θ( 1 )
θ(L)
yt.Here,θ(L)= 1 +θ 1 L+θ 2 L^2 is the reduced form MA polynomial of the local linear
trend model (9.2). The numerator of the filtered detrended series can be rewritten:
θ(L)−θ( 1 )=θ∗( 1 )L+^2 θ 0 ∗, withθ∗(L)=θ 0 ∗+θ∗ 1 L=−(θ 1 +θ 2 )−θ 2 L.
The expression forψ ̃t|∞is sometimes mistakenly taken to imply that the Leser–
HP cycle filter makes stationary series that are integrated up to the fourth order,
due to the presence of| 1 −L|^4 =( 1 −L)^2 ( 1 −L−^1 )^2 in the numerator of the filter.
It should be recalled that the above formula holds true only for a doubly-infinite
sample, and the real-time filter for extractingψ ̃t|tcontains only the factor^2.
9.7 Appendix C: State-space models and methods
The models considered in this chapter admit the state-space representation:
yt=Ztαt+Gtt, t=1, 2,...,n,
αt=Ttαt− 1 +Htηt, (9.31)wheret∼NID( 0 ,I),ηt∼NID( 0 ,I), and E(tη′t)= 0. The initial conditions are
specified as follows:α 0 =α ̃∗ 0 | 0 +W 0 δ+H 0 η 0 , so thatα 1 |δ∼N(α ̃∗ 1 | 0 +W 1 δ,P∗ 1 | 0 ),
whereα ̃ 1 ∗| 0 =T 1 α ̃∗ 0 | 0 ,W 1 =T 1 W 0 , andP∗ 1 | 0 =H 1 H′ 1 +T 1 H 0 H′ 0 T 1 ′. The random
vectorδcaptures the initial conditions for non-stationary state components and is
assumed to have a diffuse distribution,δ∼N( 0 ,δ), withδ−^1 →0. The matrices
Zt,Gt,Tt,Ht,W 0 are deterministically related to a set of hyperparameters,.
For instance, for the bivariate model of output and inflation considered in
section 9.3.1,ytis a bivariate time series,αt=(μt,βt,ψt,ψt− 1 ,τt)′,Zt=Z=
(zy,zp)′,z′y = (1, 0, 1, 0, 0),z′p = (0, 0, 0, 0, 1),t = εt/σε,Gt = G = (0,σε)′,
ηt=(ηt/ση,κt/σκ,υt/συ)′,δ=(μ 0 ,β 0 ,τ 0 )′,α ̃∗ 0 | 0 = 0 ,
Tt=T=⎛
⎜
⎝Tμ 00
0Tψ 0
0 ′ t′p 1⎞
⎟
⎠,Tμ=(
11
01)
,Tψ=(
φ 1 φ 2
10)
,tp=(
θτ 0 φ 1 +θτ 1
θτ 0 φ 2)
,Ht=H=⎛
⎜⎜
⎜
⎜⎜
⎝ση 00
000
0 σκ 0
000
0 θτ 0 σκ 0⎞
⎟⎟
⎟
⎟⎟
⎠,W 0 =⎛
⎜⎜
⎜
⎜⎜
⎝100
010
000
000
001⎞
⎟⎟
⎟
⎟⎟
⎠,η 0 ∼N( 0 ,I 2 ),H 0 =⎛
⎜
⎝0
Cψ
0⎞
⎟
⎠,whereCψis such that E(ψ 0 ψ′ 0 )=CψC′ψ,ψ 0 =(ψ 0 ,ψ− 1 )′.