400 Structural Time Series Models
The first expression shows that the weights for the extraction of the cycle sum
to zero. Sinceφ( 1 )θ(L)−θ( 1 )φ(L)must have a unit root, we can writeφ( 1 )θ(L)−
θ( 1 )φ(L)=θ(L), and substituting this into (9.13), the ARMA representation for
this component can be established asφ(L)ct=θ(L)[φ( 1 )]−^1 ξt. As the order ofθ(L)
is max(p,q)−1, the cyclical component has a stationary ARMA(p, max(p,q)− 1 )
representation. For the ARIMA(2,1,2) model fitted to US GDP, the cycle has the
ARMA(2,1) representation:
φ(L)ct=( 1 +θL)[
1 −
θ( 1 )
φ( 1 )]
ξt, θ=−
φ 2 θ( 1 )+θ 2 φ( 1 )
φ( 1 )−θ( 1 ). (9.14)
It is apparent that the two components are driven by the innovations,ξt; the
fractionθ( 1 )/φ( 1 ), known aspersistence, is integrated in the trend, and its comple-
ment to 1 drives the cycle. The sign of the correlation between the trend and the
cycle disturbances is provided by the sign ofφ( 1 )−θ( 1 ); when persistence is less
(greater) than one then the trend and cycle disturbances are positively (negatively)
and perfectly correlated.
9.2.6 Model-based bandpass filters
As we said before, macroeconomic time series such as GDP do not usually admit
the decompositionyt =μt+ψt, withψtbeing a purely irregular component;
nevertheless, applications of the class of filters (9.6) is widespread, as the popu-
larity of the Hodrick–Prescott filter testifies. However, when the available series
ytcannot be modeled as (9.2), it is not immediately clear how the components
should be defined and how inferences about them should be made. In particular,
the Kalman filter and the associated smoothing algorithms no longer provide the
minimum mean square estimators of the components nor their mean square error.
The discussion of model-based bandpass filtering in a more general setting will be
the theme of this section.
The trend-cycle decompositions dealt with in the two previous sections are mod-
els of economic fluctuations, such that the components are driven by random
disturbances which are propagated according to a transmission mechanism. In
this section we start from a reduced form model (as in the case of the BN decom-
position) and define parametric trend-cycle decompositions that are less loaded
with structural interpretation, since they just represent the lowpass and highpass
components in the series. The aim is to motivate and extend the use of signal
extraction filters of the class (9.6) to a more general and realistic setting than (9.5).
For this approach to the definition of bandpass filters see Gómez (2001) and Kaiser
and Maravall (2005). The following treatment is based on Proietti (2004).
Letytdenote a univariate time series with ARIMA(p,d,q) representation, which
we write as:
φ(L)(dyt−β)=θ(L)ξt, ξt∼NID(0,σ^2 ),
whereβis a constant,φ(L)= 1 −φ 1 L−···−φpLpis the AR polynomial with
stationary roots, andθ(L)= 1 +θ 1 L+···+θqLqis invertible. We are going to exploit
the fundamental idea that we can uniquely decompose the WN disturbanceξtinto