402 Structural Time Series Models
parameterλ 1 andλ 2 , determined according to (9.7). Obviouslyλ 1 >λ 2. The trend-
cycle decomposition corresponding to the triplem,r,λ 2 (or, equivalently,m,r,ωc 2 ),
is as in (9.17):
yt=μ 2 t+ (^) t,
dμ 2 t=β+
( 1 +L)r
φ 2 (L)
θ(L)
φ(L)
ζ 2 t, ζ 2 t∼NID(0,σ^2 ) (9.18)
(^) t=
( 1 −L)m
φ 2 (L)
θ(L)
dφ(L)
κ 2 t, κ 2 t∼NID(0,λ 2 σ^2 ),
with|φ 2 (L)|^2 =| 1 +L|^2 r+λ 2 | 1 −L|^2 m.
We can similarly define the trend-cycle decomposition corresponding to the
triplem,r,λ 1 (or, equivalently,m,r,ωc 1 ),yt=μ 1 t+ψt.Asλ 1 >λ 2 this decompo-
sition features a lower cut-off frequency,ωc 1 , thereby yielding a smoother trend.
The componentsμ 1 tandψtare defined as in (9.18), withφ 1 (L),ζ 1 t∼NID(0,σ^2 )
andκ 1 t∼NID(0,λ 1 σ^2 )replacing, respectively,φ 2 (L),ζ 2 tandκ 2 t. The polynomial
φ 1 (L)is such that|φ 1 (L)|^2 =| 1 +L|^2 r+λ 1 | 1 −L|^2 m.
The lowpass component,μ 2 t, can, in turn, be decomposed using the orthogonal
decomposition of the disturbanceζ 2 t:
ζ 2 t=
φ 2 (L)
φ 1 (L)
ζ 1 t+
( 1 −L)m
φ 1 (L)
κ 1 t, (9.19)
with:
ζ 1 t∼NID(0,σ^2 ), κ 1 t∼NID
(
0,(λ 1 −λ 2 )σ^2
)
,E(ζ 1 jκ 1 t)=0, ∀j,t.
Under this setting, the spectrum of both sides of (9.19) is constant and equal to
σ^2 / 2 π.
Substituting (9.19) into (9.18), and writingμ 2 t =μ 1 t+ψt, enablesytto be
decomposed into three components, representing the lowpass(μ 1 t), bandpass(ψt)
and highpass(
t)components, respectively.
yt=μ 1 t+ψt+ (^) t,
dμ 1 t=c+(^1 +L)
r
φ 1 (L)
θ(L)
φ(L)ζ^1 t, ζ^1 t∼NID(0,σ
(^2) )
ψt=(^1 +L)
n( 1 −L)m
φ 1 (L)φ 2 (L)
θ(L)
dφ(L)κ^1 t, κ^1 t∼NID
(
0,(λ 1 −λ 2 )σ^2 )
)
,
(9.20)
and (^) t, given in (9.18), is the highpass component of the decomposition (9.20).
The model can be cast in state-space form and the Kalman filter and smoother
(see Appendix C) will provide the optimal estimates of the components and their
standard errors.
Figure 9.3 shows the gain of an ideal bandpass filter and the BK filter. The dashed
line is the gain of the model-based bandpass filter which is optimal forψtin (9.20)