Stephen G. Hall and James Mitchell 225is essentially nonparametric and can accommodate any possible distribution. For
finiteNthe mixture distribution still offers a very flexible modeling approach.
Further characteristics of the combined densityp(yt|t−h)can be drawn out by
definingmitandvitas the mean and variance of forecasti’s distribution at timet:
mit=∞∫−∞ytg(yt|it−h)dytandvit=∫∞−∞(yt−mit)^2 g(yt|it−h)dyt;(i=1,...,N).Then the mean and variance of (5.40) are given by:^7
E[
p(yt|t−h)]
=∑Ni= 1wimit, (5.42)Var[
p(yt|t−h)]
=∑Ni= 1wivit+∑Ni= 1wi{
mit−m∗t} 2. (5.43)
(5.43) indicates that the variance of the combined distribution equals average
individual uncertainty (“within” model variance) plus disagreement (“between”
model variance).^8 This result stands in contrast to that obtained when combining
point forecasts, where combination using “optimal” (variance or RMSE minimiz-
ing) weights means the RMSE of the combined forecast must be equal to or less
than that of the smallest individual forecast (see Bates and Granger, 1969, and, for
related discussion in a regression context, Granger and Ramanathan, 1984). Den-
sity forecast combination will in general increase the combined variance. However,
this increase in uncertainty need not be deleterious; when evaluated the com-
bined density forecast may perform better than the individual density forecasts.
Hall and Mitchell (2004) distinguish between combining competing forecasts of
various moments of the forecast density and directly combining the individual
densities themselves, as with the finite mixture density.
Focusing on the predictive accuracy of the combination, rather than the indi-
vidual components, the key practical issue is to determinewi.^9 We consider two
methods in sections 5.5.3 and 5.5.4.
5.5.3 Equal weights
Most simply, equal weights,wi= 1 /N, have been advocated (see Hendry and
Clements, 2004; Smith and Wallis, 2008). Indeed, equal weights are used by the SPF
when publishing their combined density forecasts. Also based on equal weights,
there are derivative combination methods which use somead hocrule, such as
trimming or thick-modeling (Granger and Jeon, 2004), to eliminate thek% worst
performing forecasts and then take an equal weighted average of the remaining
forecasts.
As experience of combining point forecasts has taught us, irrespective of its
performance in practice, use of equal weights is only one of many options. For
example, one popular alternative to equal weights in the point forecast literature,