Palgrave Handbook of Econometrics: Applied Econometrics

(Grace) #1

208 Recent Developments in Density Forecasting


Finally the reflecting barrier alters the transition density to give:

p(θt+ 1 |θt,Q)∝I(θt+ 1 )f(θt+ 1 |θt,Q)


I(θt+2,T)f(θt+2,T|θt+ 1 ,Q)dθt+2,T.
(5.13)

So far this defines a VAR with parameters which follow a random walk with the
additional constraint that the parameters are not allowed to wander into a region
with unstable behavior. In addition to this, Cogley, Morozov and Sargent (2005)
further extend the model to include a drifting conditional variance. Following the
stochastic volatility literature they define:


Rt=B−^1 HtB−^1


, (5.14)

whereBis lower triangular with unity along the main diagonal andHis assumed
diagonal with univariate stochastic volatilities along the main diagonal which
evolve as:
lnhit=lnhit− 1 +σiηit, (5.15)


where theηitare mutually independent volatility innovations andσiis a free
parameter.
This model then generates a very rich density function as it involves both
changing parameters in the VAR and an error term which follows a time-varying
stochastic distribution. In terms of a pure model-based density forecast, this type of
Bayesian framework is probably as general as is currently possible. Cogley, Morosov
and Sargent (2005) detail how to construct the fan chart by simulating the BVAR
posterior predictive density,p(YT+1,T+F|YT).


5.3.3 Subjective density forecasts


Most of the density forecasts which are produced regularly by national or interna-
tional institutions are, however, constructed in a much less formal way. The SPF,
mentioned in the introduction to this section, is largely judgmental. Each fore-
caster arrives at his or her own forecast in completely different ways; some may
use models but most certainly do not. The key questions of interest here focus
on each individual’s view of the likely uncertainty surrounding their forecast for
inflation and output growth. These individual forecasts are then presented as a set
of histograms which are then averaged, using equal weights (see section 5.5), to
give a mean density forecast.
The Bank of England began publishing density forecasts at the beginning of 1993.
At the beginning of 1996 the Bank changed its methodology somewhat. Before the
last quarter of 1995 the Bank’s density forecast was implicitly normal. Since the
beginning of 1996 the Bank of England has stated clearly that its density forecasts
are non-normal, following a two-piece normal distribution; i.e., each side of the
mode has a normal shape but they do not have the same standard deviation –
hence each side does not represent half of the distribution. This distribution is
arrived at as the subjective assessment by the Bank’s Monetary Policy Committee
(MPC), based partly on past forecast errors, partly on a range of formal models, and
partly on subjective judgments regarding the asymmetry of risk in the forecast.

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