Stephen G. Hall and James Mitchell 205is the contribution coming from incorrect exogenous assumptions, and(Y^4 −̂Y)is
the contribution to the complete forecast error coming from the judgment imposed
on the forecast from outside the model. This is a useful way of categorizing the total
error composition; however, numerically the order in which this decomposition
is carried out can affect the numerical evaluation of these components (see Hall
and Henry, 1988). It is also worth noting that the variances which lie behind
these components and which make up the complete density of the forecast may
be either time varying or constant over time, and there will generally be non-zero
covariances between these components.
Part of the usefulness of this decomposition is to emphasize the range of sources
of uncertainty. Virtually no formal model-based analysis can deal with all of these
and so we may justify the use of judgment, which will be discussed below, at least
partly on the grounds of this failure on the part of the formal analysis.
5.3.2 Model-based densities
If we now turn to density forecasts produced by a range of models, a natural starting
place is a conventional VAR, as this may be thought to capture the basic properties
of most standard forecasting models, including large macroeconomic forecasting
models. Indeed, the linearized solution of DSGE models, the workhorse of mod-
ern macroeconomics (see Woodford, 2003) and from which density forecasts may
readily be constructed using simulation methods, can, under certain conditions, be
approximated by a (restricted) finite-order VAR (e.g., see Pesaran and Smith, 2006;
Ravenna, 2007).
Thus, consider a VAR of the form:
Yt=B(L)Yt− 1 +εt, (5.3)(t=1,...,T), whereYtis a vector ofNvariables,B(L)is a suitably dimensioned
matrix lag polynomial of estimated parameters with fixed covariance matrix and
εtis anN×1 vector of residuals with constant covariance matrix. Given the con-
stant covariance assumption for both the parameters and residuals, the density
of a forecast ofYt+h(h=1,...,H) will, in general, vary only with the initial
valuesYt− 1. If the errors in the VAR process are normally distributed then the den-
sity function of the VAR forecasthsteps ahead is also normally distributed with
a covariance structure which can be approximated analytically. Lutkepohl (1991,
p. 87) provides an approximate analytical expression for the conditional variance
equal to the approximate mean squared error of the forecast with parameter uncer-
tainty. Allowance can also be made for the uncertain parameters of the VAR or
non-normality by using either Monte Carlo methods or bootstrap techniques (see
Garrattet al., 2006, Ch. 7). For a Bayesian approach, see Zellner (1971, pp. 233–6).
In most practical settings, the value ofYt− 1 will not vary sufficiently to produce
large variations in the shape of the density ofYt+h, and so for most practical pur-
poses we can assume that the density is constant across time except for the mean
of the distribution. Hence there is little interest in regularly publishing full details
of the density forecast from this type of model, as the only element of the density
which would change substantially through time is the mean.