232 Recent Developments in Density Forecasting
VAR model forecast densities, as measured by the logarithmic score, over the hold-
out period. This weighted combination gives greater weight to models that allow
for the shifts in volatilities associated with the Great Moderation. This result again
contrasts with that typically found with point forecasts, where equal weighted
averages are hard to beat.
5.6 Conclusion
The past decade has seen a considerable increase in the production and use of
density forecasts in macroeconomics. This reflects both changes in the dynam-
ics of the macroeconomy, with important shifts in both the level and volatility of
many macroeconomic variables making it important to forecast the overall density
function rather than just the conditional mean, and the development of econo-
metric models that allow for time variation in the conditional variance, as well
as the conditional mean. With this increased use of density forecasts, important
new econometric challenges have arisen as macroeconomists seek recourse to a
toolbox comparable to that routinely used both to produce and use point (condi-
tional mean) forecasts. This chapter has reviewed recent additions to this toolbox,
focusing on the practicality rather than rigour of the methods.
This first involved surveying some methods for the production of density fore-
casts. We also considered combining model-based and subjective information, by
twistingthe model-based densities to reflect prior (perhaps subjectively formed)
information. We should imagine that the techniques considered could be of par-
ticular use to professional forecasters, like many central banks, who use both
model-based information and judgment when forming their density forecasts. Sec-
ond, we provided a practical discussion of methods for theex postevaluation of
density forecasts. This involved discussion of both rolling and fixed-event density
forecasts. Numerous tests for both absolute and relative density forecasting per-
formance, using both the probability integral transforms and scoring rules, were
discussed, and their relationship to the Kullback–Leibler information criterion con-
sidered. But we stressed the need for further work to establish a consensus on
the appropriate test(s), especially in the small-samples typical of macroeconomics
and when forecasting more than one step ahead. Finally, again reflecting a com-
mon situation for many macroeconomists who forecast from a suite of models,
we reviewed methods for the combination of density forecasts to overcome the
uncertainty in model selection. Particular focus was given to how to choose the
combination weights. In contrast to the conventional wisdom about point fore-
casts, where equal weights are generally preferred, recent applied work has found
that the predictive accuracy of combined density forecasts improves when a greater
weight is given to models that allow for the shifts in volatility which have been
observed in many economies over the last 20 years.
Acknowledgments
The authors have benefited from numerous discussions with Shaun Vahey and Ken Wallis
and thank John Geweke and Terry Mills for useful comments.