Palgrave Handbook of Econometrics: Applied Econometrics

224 Recent Developments in Density Forecasting

The “linear opinion pool” takes a weighted linear combination of the forecasters’
probabilities. The combined density is then defined as the finite mixture:

p(yt|t−h)=

∑N

i= 1

wig(yt|it−h), (5.40)

whereg(yt|it−h)are theh-step-ahead density forecasts of modeli(i=1,...,N)
of a random variableytat timet(t=1,...,T) conditional on the information

setit−h, andt−h=∪Ni= 1

{

it−h

}

. The set of non-negative weights,wi, sum to
unity. The restriction that each weight is positive might be relaxed (for discussion
and references, see Genest and Zidek, 1986). In the finite mixture distribution
the weights, the mixing proportions, are positive by construction (see Everitt and
Hand, 1981). In (5.40) the weights are assumed time-invariant,wit=wi, since
below we consider their estimation using sample averages (t=1,...,T). But in
general, e.g., when computed on an out-of-sample (recursive) basis, they can be
time-varying. (5.40) satisfies certain properties such as the “unanimity” property
(if all forecasters agree on a probability then the combined probability agrees also).
For further discussion, and consideration of other properties, see Genest and Zidek
(1986) and Clemen and Winkler (1999).
The logarithmic opinion pool is defined as:

p(yt|t−h)=k

∏N

i= 1

g(yt|it−h)wi, (5.41)

wherekis a normalizing constant. Whenwi=( 1 /N),p(yt|t−h)is proportional
to the geometric mean of the experts’ distributions. In (5.41)p(yt|t−h)is, in fact,
that density forecast “closest,” in a KLIC sense, to each of theNcompeting density
forecasts (see Heskes, 1998).

5.5.2 The linear opinion pool

We follow Mitchell and Hall (2005), Wallis (2005), Timmermann (2006, p. 177)
and Hall and Mitchell (2007) and focus on density forecast combination via the
linear opinion pool. Indeed, (5.40) offers a well understood and much exploited
means of combining density forecasts. The SPF, previously the ASA-NBER survey,
has essentially used it since 1968 to publish a combined density forecast of infla-
tion, amongst other things, from the individual-level density forecasts which are
supplied to it.
Inspection of (5.40) reveals that taking a weighted linear combination of the fore-
casters’ densities can generate a combined density with characteristics quite distinct
from those of the forecaster, although this will not be the case for the combination
of natural-conjugate densities (Winkler, 1968). For example, if all the forecasters’
densities are normal, but with different means and variances, then the combined
density will be mixture normal. Mixture normal distributions can have heavier
tails than normal distributions, and can therefore potentially accommodate skew-
ness and kurtosis. Combining individual normal density forecasts may mitigate
misspecification of the individual densities. AsN→∞the mixture distribution