Stephen G. Hall and James Mitchell 231gains in forecast accuracy when some variant of KLIC minimizing or BMA, rather
than equal, weights are used. This contrasts with the conventional wisdom about
point (conditional mean) forecasts, where equal weights are generally preferred.
However, for other samples and sets of models, Gerard and Nimark (2008) and
Kascha and Ravazzolo (2008) have found that equal weight density combinations
can remain hard to beat.
Hall and Mitchell (2007) combine, based on their out-of-sample fit, the one-
year-ahead density forecasts of UK inflation published in real time by the Bank
of England and the NIESR; they search for the set of weights that maximize the
logarithmic score, cf. (5.46). When this is undertaken over the full sample period
the combined density forecast is found not to beat the best individual density
forecast, the Bank’s forecast. The KLIC minimizing weights are unity on the Bank
and zero on the NIESR, i.e., there is no combination. This is consistent with the
view that combination with an inferior forecast need not help. But, similar to when
variance minimizing weights are used to combine point forecasts, combination
does not make matters worse (as judged by the logarithmic score, and for point
forecasts as judged by the RMSE) when the weights are selected using full-sample
information (i.e., in-sample). But when the weights are chosen recursively (i.e.,
out-of-sample) this need not be the case. Hall and Mitchell (2007) find that an
equal weighted combination produces a less accurate density forecast.
Again combining the Bank’s and the NIESR’s inflation densities, Mitchell and
Hall (2005) consider the merits of an alternative means of deriving the combina-
tion weights. This involves using−i=KLICit−h−min(KLICit−h)in (5.50), with
KLIC
i
t−hestimated fromLR
i
Bwith two degrees-of-freedom. Mitchell and Hall (2005)
then find, even in-sample, that the weighted combination performs worse than the
Bank’s density, as measured by both the logarithmic score and theLRiBtest statistic.
This can be attributed to the danger, discussed in Hall and Mitchell (2007), that
the true density forz∗it|t−h, essential to estimation ofKLIC
i
t−h, need not be nested
by the chosen specification forq(.). Parameter uncertainties in small samples, due
to the estimation of, in this case,μandσ, may also be playing a role. Reconciling
and comparing the efficacy of the alternative methods of estimatingw∗iandwiBMA
remains the subject of ongoing research.
The starting point for Jore, Mitchell and Vahey (2008) is the application of Clark
and McCracken (2008) to real-time US data. Clark and McCracken (2008) argue
that combining real-time point forecasts from VAR models of output, prices and
interest rates improves point forecast accuracy in the presence of uncertain model
instabilities. Using the same real-time dataset, Jore, Mitchell and Vahey (2008) gen-
eralize their approach to consider forecast density combinations and evaluations.
Whereas Clark and McCracken (2008) show that the point forecast errors from par-
ticular equal-weight pairwise averages are typically comparable to or better than
benchmark univariate time series models, Jore, Mitchell and Vahey (2008) show
that neither approach produces accurate real-time forecast densities for recent US
data. But substantially improved predictive density accuracy is obtained when the
competing density forecasts are combined on the basis of the fit of the individual