422 Structural Time Series Models
and Rossi (2004) and Proietti, Musso and Westermann (2007). The implications
of the uncertainty surrounding the output gap estimates for monetary policy are
considered in Orphanideset al. (2000) and Ehrmann and Smets (2003), among
others.
A full assessment of the output gap reliability is complicated by the very nature
of the measurement which, like the NAIRU, core inflation, and so forth, refers to
a latent variable, for which there is no underlying “true value” to be elicited by
other data collection techniques.
The previous sections have presented different parametric methods that can be
used to measure the underlying signals. Assume that there is a true output gapψt
and that there is an approximating model, denoted byM, providing a represen-
tation for it. The model specifies how the observations are related to the object of
the measurement. Let us denote byψt,Mthis (parametric) representation. Now, let
ψ ̃t,Mdenote the estimator ofψtbased on modelM, i.e., using the representation
ψt,M. We assume thatψ ̃t,Mis the optimal signal extraction method forψt,M.
How do we judge the reliability ofψ ̃t,M? Reliability is a statement concerning the
closeness ofψ ̃t,Mandψt. Following Boumans (2007), two key features are accu-
racy and precision, as the discrepancyψ ̃t,M−ψtcan be broken down into two
components:(ψ ̃t,M−ψt,M)+(ψt,M−ψt), which are associated respectively to
the precision of the method, and to the accuracy or validity of the representation
chosen. Given the information setF, precision is measured by (the inverse of)
Var(ψt,M|F)=E[(ψ ̃t,M−ψ ̃t,M)^2 |F].
9.4.1 Validity
Validity is usually difficult to ascertain, as it is related to the appropriateness of
ψt,Mas a model for the signalψt. This is a complex assessment, involving many
subjective elements, such as any prior available information and the original moti-
vation for signal extraction. The issue is indissolubly entwined with the nature
ofψt: the previous paragraphs have considered two main perspectives. The first
regardsψtas the component of the series that results from the transmission mech-
anism of demand or nominal shocks. The second view considersψtas the bandpass
component of output.
Recently, there has been a surge of interest in model uncertainty and in model
averaging. The individual estimates ψ ̃t,Mi,i = 1, 2,...,K, may be combined
linearly, givingψ ̃t=
∑
iciψ ̃t,Mi, where the coefficientsciare proportional to the
precision of the methods, or the posterior probability in a Bayesian setting.
It is more viable to assess two other aspects of validity, namely concurrent and
predictive validity. The first is concerned with the contemporaneous relation-
ship between the measureψ ̃t,Mand a related alternative measure of the same
phenomenon. Such measures are rarely available. Although business surveys are
implemented with the objective of collecting informed opinions on latent vari-
ables, such as the state of the business cycle, they can hardly be considered as
providing a measure of the “true” underlying state of the economy.
Predictive validity relates to the ability to forecast future realizations ofytor
related variables; evaluating the mean forecast error yields useful insight on its