Palgrave Handbook of Econometrics: Applied Econometrics

D.S.G. Pollock 275

the canonical decomposition of the generating function can be represented by:

θ(z)θ(z−^1 )

φ(z)φ(z−^1 )

=

γT(z)γT(z−^1 )

φT(z)φT(z−^1 )

+

γS(z)γS(z−^1 )

φS(z)φS(z−^1 )

+ρ(z)ρ(z−^1 ). (6.103)

There are now two improper rational functions on the right-hand side, which have
equal degrees in their numerators and denominators.
According to Wiener–Kolmogorov theory, the optimal signal-extraction filter for
the trend component is:

βT(z)=

γT(z)γT(z−^1 )

φT(z)φTz−^1 )

×

φS(z)φT(z)φT(z−^1 )φS(z−^1 )

θ(z)θ(z−^1 )

=

γT(z)γT(z−^1 )φS(z)φS(z−^1 )

θ(z)θ(z−^1 )

=

CT(z)

θ(z)θ(z−^1 )

.

(6.104)

This has the form of the ratio of the autocovariance generating function of the trend
component to the autocovariance generating function of the processy(t). This
formulation presupposes a doubly-infinite data sequence, so it must be translated
into a form that can be implemented with finite sequences.
The approach to the estimation of unobserved components that adopts the prin-
ciple of canonical decompositions has been advocated by Hillmer and Tiao (1982)
and Maravall and Pierce (1987). It has been implemented in the TRAMO–SEATS
program of Gómez and Maravall (1996) and Caporello and Maravall (2004), which
builds upon the work of Burman (1980).

6.6.4 The state-space form of the structural model

In the foregoing approach to modeling the components of a structural time series
model, an aggregate univariate process is first estimated and then decomposed
into its components. An alternative approach is to model the individual compo-
nents from the start as separate entities, which are described by independent linear
stochastic models.
Provision can be made for a cyclical component which is distinct from the trend
component, but, if this is omitted, then the disaggregated model commonly takes
the form ofy(z)=τ(z)+σ(z)+η(z), where:

τ(z)=

( 1 +αz)

∇^2 (z)

ζ(z), (6.105)

σ(z)=

1

S(z)

ω(z). (6.106)

τ(t)is the trend,σ(t)is the seasonal component andη(t)is the irregular noise.
Here there are three independent white-noise processes driving the model, which
areζ(t),ω(t)andη(t). The model has been described by Harvey (1989) as the basic
structural model. A reason for omitting the cyclical or business-cycle component
from this model is the difficulty in separating it from the trend component.