Tommaso Proietti 387The signal extraction problems relating to latent variables, such as the output
gap, core inflation and the NAIRU, can be consistently formulated within a model-
based framework and, in particular, within the class of unobserved components
time series models, so matching the fundamental economic relationships with
observable macroeconomic aggregates.
The chapter is divided into three main parts: the first (section 9.2) deals with
univariate methods for cycle measurement. One approach is to formalize a model
of economic fluctuations such that the different components are driven by specific
shocks that are propagated via a dynamic transmission mechanism. We start by
introducing the traditional trend-cycle structural decomposition, discussing the
parametric representation of both components (sections 9.2.1–4) and the correla-
tion between the trend and cycle disturbances (section 9.2.5). Another approach is
to consider the cycle as the bandpass component of output, i.e., as those economic
fluctuations which have a periodicity greater then a year and smaller than, say,
eight years. We review the relationship between popular signal extraction filters,
such as the Hodrick–Prescott and the Baxter and King filters, and the model-based
Wiener–Kolmogorov filter. Particular attention is devoted to the implementation
of bandpass filtering in a model-based framework (section 9.2.6). The advantages
of this strategy are twofold: the components can be computed in real time using
standard principles of optimal signal extraction, so that efficient algorithms, such
as the Kalman filter and smoother, can be applied. Second, the reliability of the
estimated components can be thoroughly assessed.
The second part, starting with section 9.3, deals with multivariate models for the
measurement of the output gap. The above definition of the output gap as an indi-
cator of inflationary pressures suggests that the minimal measurement framework
is of a bivariate model for output and inflation. After reviewing the work done in
this area (section 9.3.1), we illustrate the estimation of a bivariate model for the
US economy, under both the classical and the Bayesian approaches, and incorpo-
rating the feature known as the “Great Moderation” of the volatility of economic
fluctuations (section 9.3.2). In section 9.3.3 we review the multivariate extensions
of the basic bivariate model and we conclude this part with an application which
serves to illustrate the flexibility of the state-space methodology in accommodat-
ing data features such as missing data, nonlinearities and temporal aggregation.
In particular, section 9.3.4 presents the results of fitting a four-variate monthly
time series model for the US economy with mixed frequency data, as gross domes-
tic product (GDP) is available only quarterly, whereas industrial production, the
unemployment rate and inflation are monthly. The model incorporates the tem-
poral aggregation constraints (which are nonlinear since the model is formulated
in terms of the logarithm of the variables) and produces as a byproduct monthly
estimates of GDP, along with their reliability, that are consistent with the quarterly
observed values.
The third part, section 9.4, deals with the reliability of the output gap
estimates. The assessment of the quality of the latter is crucial for the deci-
sion maker. We discuss the various sources of uncertainty (model selection,
parameter estimation, data revision, estimation of unboserved components,