Stephen G. Hall and James Mitchell 223- has begun to explore further why point forecast combination works through
analytical and Monte Carlo investigation. But, given that measures of uncertainty
surrounding a point forecast enhance its usefulness, the natural next step is to con-
sider density forecast combination. While Clements (2006) and Granger, White
and Kamstra (1989) have considered, respectively, the combination of event and
quantile forecasts, that inevitably involve a loss of information compared with
consideration of the “whole” density, the combination of density forecasts has
been relatively neglected. In fact, Clements (2003, p. 2) identified this as “an area
waiting investigation.”
5.5.1 Combination methods
While methods for combining point forecasts are well established and much
exploited, less direct attention has been given in econometrics to the combina-
tion of density forecasts. This is also a concern in practice since many professional
forecasters, particularly central banks, consult more than one forecast. Central
bankers often look at what they call a “suite of models.” These competing fore-
casts are produced using both structural macro (usually large-scale but increasingly
DSGE) models and atheoretical models, as well as variants in-between. In addition,
central bankers routinely add off-model information (“judgment”) to model-based
forecasts to produce predictive densities. The issue then again arises as to how
they should internally reconcile or combine competing density forecasts of the
same event to arrive at a single density which is then used to communicate
policy.
However, the expert combination literature, more commonly seen in manage-
ment science and risk analysis journals, has considered density forecast combina-
tion, although not evaluation as discussed in section 5.4. This literature adopts
a Bayesian approach whereby competing densities are combined by a “decision
maker” who views them as data that are used to update a prior distribution (for
reviews see Genest and Zidek, 1986; Clemen and Winkler, 1999). There is also, as
we discuss in section 5.5.4.1, a related Bayesian literature in econometrics, but only
recently has this turned to the combination and evaluation of combined density
forecasts. Its use for the combination of subjectively-formed density forecasts is
also little discussed.
Within the expert combination literature Clemen and Winkler (1999) distin-
guish behavioral and mathematical approaches to combination. The behavioral
approach seeks to combine experts’ opinions by letting the experts interact in some
manner to reach a collective opinion. This approach is not considered further since
one can imagine many situations in economic forecasting, e.g., when forecasts
are model-based, when it is inappropriate. By contrast, mathematical approaches
combine the information across experts by using some rule or model. Work has
focused on combination rules that satisfy certain properties or axioms. Two com-
mon axiomatic approaches are the “linear opinion pool” (Morris, 1974, 1977;
Winkler, 1981; Lindley, 1983; Genest and McConway, 1990) and the “logarithmic
opinion pool.”