Palgrave Handbook of Econometrics: Applied Econometrics

276 Investigating Economic Trends and Cycles

The trend process is usually depicted as the product of two processes that
constitute the so-called local linear model, which has already been described in
section 6.5.1:

τ(t)=τ(t− 1 )+β(t)+ν(t), (6.107)

β(t)=β(t− 1 )+ε(t). (6.108)

The first of these describes the level of the trend process and the second describes
its slope.
A more elaborate seasonal model is available that generates more regular cycles.
A moving-average operatorM(z)can be included in the numerator of the expres-
sion on the right-hand side of (6.106) to give σ(z) ={M(z)/S(z)}ω(z). The

autoregressive operator may be factorized asS(z)=

∏s− 1

j= 1 (^1 −e

2 πj/s), wheresis the

number of observations per annum. The complementary moving-average opera-

tor will have the form ofM(z)=

∏s− 1

j= 1 (^1 −ρe

2 πj/s), whereρ<1 is close to unity.

The zeros of the moving-average operator will serve largely to negate the effects of
the poles of the autoregressive operator, except at the seasonal frequencies, where
prominent spectral spikes will be found.
The basic structural model, without the elaboration of a seasonal moving-average
component, can be represented in a state-space form that comprises a transi-
tion equation, which describes a first-order vector autoregressive process, and an
accompanying measurement equation. For notational convenience, lets=4,
which corresponds to the case of quarterly observations on annual data. Then
the transition equation, which gathers together equations (6.106), (6.107) and
(6.108), is:
⎡
⎢⎢
⎢⎢
⎢
⎣

τ(t)

β(t)

σ(t)

σ(t− 1 )

σ(t− 2 )

⎤

⎥⎥

⎥⎥

⎥

⎦

=

⎡

⎢⎢

⎢⎢

⎢

⎣

11000

01000

00 − 1 − 1 − 1

00100

00010

⎤

⎥⎥

⎥⎥

⎥

⎦

⎡

⎢⎢

⎢⎢

⎢

⎣

τ(t− 1 )

β(t− 1 )

σ(t− 1 )

σ(t− 2 )

σ(t− 3 )

⎤

⎥⎥

⎥⎥

⎥

⎦

+

⎡

⎢⎢

⎢⎢

⎢

⎣

ν(t)

ε(t)

ω(t)

0

0

⎤

⎥⎥

⎥⎥

⎥

⎦

. (6.109)

This incorporates thetransitionequation of the non-seasonal local linear model
that has been given by (6.64). The observation equation, which combines the
current values of the components, is:

y(t)=

[

10100

]

⎡

⎢

⎢⎢

⎢

⎢

⎣

τ(t)

β(t)

σ(t)

σ(t− 1 )

σ(t− 2 )

⎤

⎥

⎥⎥

⎥

⎥

⎦

+η(t). (6.110)

The state-space model is amenable to the Kalman filter and the associated smooth-
ing algorithms, which can be used in estimating the parameters of the model and
in extracting estimates of the so-called unobserved componentsτ(t),σ(t)andε(t).
These algorithms have been described by Pollock (2003a).