Stephen G. Hall and James Mitchell 211(KLIC), which may be stated as:
K(π∗:π)=∑N
i= 1 π∗
ilog(
π∗i
πi). (5.19)
Obviously, ifπi∗=πithen the KLIC will be zero, so this is in effect measuring
the distance between the two distributions. Ifπiis uniform then this is sometimes
termed the entropy; if it is non-uniform it is termed the relative entropy.
The idea is simply to choose a set of weightsπ∗which minimizeK(π∗:π)
subject to the following set of restrictions:
πi∗≥0;∑N
i= 1
πi∗=1;∑N
i= 1
πi∗g(Yi)=g. (5.20)Once we have solved for the new weights we may then easily calculate the new
density forecast simply by weighting together all the original draws with the new
weightsπ∗.
Robertson, Tallman and Whiteman (2005) illustrate this technique by consid-
ering a small VAR model for the Federal Funds rate, inflation and the output gap
and then imposing a set of prior restrictions on the forecast via a set of moment
conditions. These include the prior view that inflation should be at its target rate
of 2.5% three years in the future, various assumptions regarding the operation of
a Taylor rule, and that the output gap eventually closes. They argue that there is
some evidence that the restricted forecasts are an improvement over the standard
VAR.
Cogley, Morosov and Sargent (2005) combine this technique with their time-
varying parameter BVAR with stochastic volatility, outlined above, to generate
forecasts for UK inflation which they contrast with the Bank of England’s den-
sity forecast. They find that to impose the Bank of England’s density forecast on
the VAR forecasts requires a considerable change to the weighting vector, some-
times referred to as “twisting” the weights. They warn that the relative entropy
(KLIC distance) points to “a severe twisting,” which may be interpreted as saying
that the Bank’s density is a long way from the BVAR model and, on this basis, they
recommend “a careful review of the evidence being used to twist the forecast.”
5.4 The evaluation of density forecasts
In practice, forecasters make successive forecasts of the same event, so-called “fixed-
event” forecasts, as well as series of forecasts of fixed lengthh, so-called “rolling”
forecasts. There exist well established statistical techniques for theex postevalu-
ation of both fixed event and rolling point forecasts. For rolling point forecasts
these are often based around the RMSE of the forecast relative to the subsequent
outturn. Indeed, publication of RMSE statistics is itself a welcome indication that
point forecasts are uncertain; in the absence of knowledge of the true loss func-
tion, squared-error loss has become the most commonly used function (see Lee,
2007, for a review of loss functions). The unbiasedness and efficiency of point fore-
casts are also tested using Mincer and Zarnowitz (1969) tests. For fixed-event point