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Measuring the Uncertainty Associated with the Output gap

We consider a simple approach to measure uncertainty that relies on a state-space
representation for the output gap estimator. Conditional on Gaussianity (of the
disturbances driving the components of the state vector) and knowledge of the covariance
matrix of the estimated state vector confidence intervals around the output gap can be
presented, and density estimates derived.¹²⁶ Other distributions could be considered
although we confine attention to the Gaussian case since this both has the advantage of
simplicity/familiarity and is sufficient to illustrate the role of forecasting uncertainty.

Let Pt|t denote the Kalman filter based variance of the output gap at time t, yt|t, using
information available up to time t; the output gap is one of the elements of the state vector
in the UC model. Then conditional on Gaussianity, the density is N(yt|t, Pt|t), from which
the 95% confidence interval around the point estimate, for example, can be extracted.
Note that we are considering filter uncertainty only and ignoring parameter uncertainty.
Accounting for parameter uncertainty should be expected to increase the width of the
confidence bands around the point estimates.

With known and constant parameters, uncertainty is greater around the real-time (strictly
the ‘filtered’ estimate) than the ‘final’ estimate. This is seen as follows. Denote the
revision between the real-time and final estimates by Rt|T = yt|T - yt|t. With known and
constant parameters (i.e. for the filtered rather than real-time estimates of Table 7.1), this
variance Pt|t’ decreases as t′ increases; Pt|t = Pt|T +Var(R^*t|T). The variance of the filtered
(or one-sided) estimate is therefore greater than that of the smoothed (or two-sided)
estimate, Pt|T, by a positive scalar equal to the variance of the revisions between the
filtered and smoothed estimates R _t∣_τ where R _t∣_τ = y_t∣_τ - y_t∣_t(ΘT )). This is the familiar
result that the variance of the filtered (forecasted) estimate is equal to the variance of the
outturn, Pt∣τ, plus the square of the bias, Var(R^*t∣τ). τhis assumes the revision process has
mean zero so that Var(R _t∣_τ)=E(R t∣_τ)².

Just as point estimates of the output gap forecast in real-time can be evaluated ex post
against the final estimates (as in τables 7.1 and 7.2), the accuracy of real-time measures
of uncertainty can and should be evaluated ex post; see Mitchell (2003) for further details.
τhis is possible since a “good” interval forecast, as defined by Christoffersen (1998),
should both have correct unconditional coverage (in the sense that on average
observations fall in the interval to the predicted degree) and secondly be such that
observations fall inside the interval in a random manner which is not clustered. Below we
focus on how the uncertainty estimates associated with the output gap can be used to
produce a measure of uncertainty associated with the cyclically adjusted budget deficit.

We detail the uncertainty concerning output gap estimates for selected years in τable 7.3.
Although recursive, real time, estimates give less ‘long term’ volatility, they are in
general noticeably less certain than full sample estimates. Hence even if we think, in real
time, that the economy has an output gap of around zero, we cannot be at all certain about
that. In 1990 six out of seven countries have real time estimates that were 30 to 100 per
cent less certain than the full sample estimate. A similar pattern is clear in 1995 as well,

τhe Kalman filter recursions automatically return estimates of the covariance matrix of the state
vector. τhe diagonal elements of these matrices then can be used to construct the confidence
intervals and density estimates. τhis approach has also been followed, for example, by Orphanides
and van Norden (2002).

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