214 Recent Developments in Density Forecasting
impliesλ=1,α=0 andβ=0 in the following variant of (5.21):
It=λpt|t− 1 +α+βt− 1 +εt, (5.22)where the indicator variableItis redefined with respect to the event forecasts.
It is also worth noting, given our reference above to the evaluation of fixed-event
probability event forecasts, that when probability event forecasts are conditionally
efficient, which they are when the density forecast from which they are extracted
is “correct” (as defined in section 5.4.2.1), we know from the law of iterated
expectations that:
E{
E(It|t−h)|t−h− 1}
=E(It|t−h− 1 ). (5.23)This implies:E(pt|t−h−pt|t−h− 1 |t−h− 1 )=0, (5.24)which says that the revision to the probability event forecast is orthogonal to infor-
mation available at(t−h− 1 ), including lagged revisions to the probability event
forecasts. Thus a testable proposition for (weakly) efficient fixed-event density fore-
casts is that revisions to probability forecasts, extracted from the density forecast,
are independent. When there is a clear objective, such as a central bank keep-
ing inflation at less than 2%, it is obvious whatato consider. However, for the
density forecast to be well calibrated overall, (5.24) needs to hold for all possible
a’s. Since an infinity of event forecasts can be extracted from the density forecast,
in an application evaluating the fixed-event aspect of the SPF density forecasts,
Mitchell (2007c) evaluates both over a large number of arbitrary events and over
events of specific interest, such as inflation falling in its “comfort zone” of 1–2%.
5.4.2 Rolling density forecasts
When evaluating the performance of density forecasts as a “whole,” economists
have tended to rely on using goodness-of-fit tests to establish whether the proba-
bility integral transforms of the forecast density with respect to the realizations of
the variable are uniform or, via a transformation, normal. In contrast, others have
employed scoring rules. Both evaluation criteria have proved popular, since they
avoid having to estimate the true but unknown conditional densityf(yt|!t−h)
(where the density of the random variableytis defined with respect to the total
information set!t−h(where the forecasters’ information sett−h⊂!t−h), and
only require a time series of realizations
{
yt}T
t= 1.(^3) We review both evaluation
criteria below.
Derivatives of both evaluation criteria have also been developed in the papers
referred to below when interest lies not in the “whole” density but in specific areas,
such as the probability of tail events or economic events of interest, such as a (one
period) recession.