388 Structural Time Series Models
statistical revision), and discuss ways of dealing with them using the state-space
methodology.
One of the objectives of this chapter is to provide an overview of the main state-
space methods and to illustrate their application and scope. The description of the
algorithms is relegated to an appendix and we refer to Harvey (1989), West and
Harrison (1997), Kitagawa and Gersch (1996), Durbin and Koopman (2001), and
the selection of readings in Harvey and Proietti (2005), for a thorough presentation
of the main ideas and methodological aspects concerning state-space methods and
unobserved components models. For the class of state-space models with Markov-
switching, see Kim and Nelson (1999b), Frühwirth-Schnatter (2006) and Cappé,
Moulines and Ryden (2005). An essential and up-to-date monograph on modeling
trends and cycles in economics is Mills (2003).
9.2 Univariate methods
In univariate analysis, the output gap can be identified as the stationary or tran-
sitory component in a measure of aggregate economic activity, such as GDP.
Estimating the output gap thus amounts todetrendingthe series and a large lit-
erature has been devoted to this very controversial issue (see, e.g., Canova, 1998;
Mills, 2003).
We shall confine our attention to the additive decomposition (after a logarith-
mic transformation) of real output,yt, into potential output,μt, and the output
gap,ψt:yt=μt+ψt. This basic representation is readily extended to handle a sea-
sonal component and other calendar components such as those associated with
trading days and moving festivals, which for certain output series, e.g., industrial
production, play a relevant role.
In the structural approach a parametric representation for the components is
needed; furthermore, the specification of the model is completed by assumptions
on the covariances among the various components. The first identifying restriction
that will be adopted throughout is thatμtis fully responsible for the non-stationary
behavior of the series, whereasψtis a transitory component.
9.2.1 The random walk plus noise model
The random walk plus noise (RWpN) model provides the most basic trend-cycle
decomposition of output, such that the trend is a random walk process with normal
and independently distributed (NID) increments, and the cycle is a pure white noise
(WN) component. The structural specification is the following:
yt=μt+ψt, t=1,...,n, ψt∼NID(0,σψ^2 ),
μt=μt− 1 +β+ηt, ηt∼NID(0,ση^2 ).(9.1)When the drift is absent, i.e., whenβ=0, the model is also known as thelocal
level model(see Harvey, 1989). We assume throughout that E(ηtψt−j)=0 for allt
andj, so that the two components are orthogonal.
Ifση^2 =0,μtis a deterministic linear trend. The one-sided Lagrange multipliertest of the null hypothesisH 0 :ση^2 =0 against the alternativeH 1 :ση^2 >0, is