204 Recent Developments in Density Forecasting
been used as a proxy for true uncertainty. As the SPF also offers a direct measure
of uncertainty, since forecasters are asked to report not just their point but den-
sity forecasts, it has provided the opportunity, not possible with the HMT dataset,
to test the reliability of disagreement as a measure of uncertainty (see Zarnowitz
and Lambros, 1987; Bomberger, 1996; Giordani and Söderlind, 2003. Boero, Smith
and Wallis, 2008, introduce a new source of survey data for the UK, the Bank of
England’s Survey of External Forecasters, which also facilitates a comparison of
disagreement and uncertainty).
Macroeconomic forecasters have also studied the density of their forecasts over a
long period, although they have not typically published, on a regular and ongoing
basis, density forecasts as such. The reason for this is, partly, that for a long time it
was felt that density forecasts were too sophisticated for the public to understand
and partly that, as the models being used made the assumption that the param-
eters and the error terms had constant covariance structures, the overall density
function would not vary from one period to another except with respect to its con-
ditional mean. There was therefore relatively little interest in publishing the same
error bands over and over again. However, it is certainly true that modelers and
forecasters in the 1970s and 1980s were calculating the uncertainty surrounding
their forecasts and, occasionally at least, publishing it. For example, the London
Business School, one of the UK’s leading forecasters at the time, began regularly
publishing the average absolute errors for its forecasts in October 1983. Some early
work in this area includes Schink (1971), Bianchi and Calzolari (1980) and Fair
(1980). Fair (1984) surveyed a range of stochastic simulation techniques which
were being used to calculate the density functions for large nonlinear forecasting
models. Hall (1986), and later Blake (1996), reported studies of the density of the
NIESR’s forecasts, again using extensive stochastic simulations.
These later model-based studies contrast strongly with the SPF, which was
purely judgmentally based. This introduces an important theme into this section,
which is the issue of combining judgment with formal model-based analysis.
Another and related theme is the production of density forecasts where the density
itself changes over time. It is only when the whole density changes in a significant
way through time that it is worth going to the lengths of publishing a regular full
density forecast.
5.3.1 Sources of uncertainty
A forecast is usually subject to a range of types of uncertainty, which we can begin
to categorize by considering the following simple decomposition.^2 LetYbe the
actual outcome of an event and let̂Ybe the forecast of that event. Then we may
decompose the forecast error into a number of components:
Y−̂Y=(Y−Y^1 )+(Y^1 −Y^2 )+(Y^2 −Y^3 )+(Y^3 −Y^4 )+(Y^4 −̂Y), (5.2)where(Y−Y^1 )is the contribution to the total error coming from the model’s
error term,(Y^1 −Y^2 )is the contribution coming from the uncertain parameters,
(Y^2 −Y^3 )is the contribution coming from misspecified functional form,(Y^3 −Y^4 )