206 Recent Developments in Density Forecasting
Given this obvious limitation of the standard model, when a researcher is inter-
ested in producing a regular model-based density forecast there is an obvious need
to base it on a model which has a richer structure that allows some interesting
time variation in the shape of the density. The earliest and most obvious model
to be developed which allowed for this possibility is the ARCH process of Engle
(1982), and the associated family of GARCH models which has grown from it. A
basic GARCH(1,1) model has the following general form:
Yt=B(L)Yt− 1 +εt (5.4)
εt=ωtht;ωt∼N(0, 1) (5.5)h^2 t=α 0 +α 1 h^2 t− 1 +α^2 ε^2 t− 1. (5.6)
Given the time variation in the variance of the error term, the complete den-
sity forecast forYt+hwill also exhibit time variation. A wide range of variants
of this basic model have grown up (for an extensive survey, see Bollerslev, Engle
and Nelson, 1994), which allow not only for time variation in the variance but
also for asymmetry and non-normality. The GARCH family of models has had an
enormous impact on econometric modeling and forecasting, especially in the area
of finance, but for the purposes being considered here it does have a number of
limitations. The first and most obvious is that there are practical difficulties in
modeling large systems of equations with GARCH-like structures. While there are
a few extensions of the GARCH approach to systems (e.g., Engle and Kroner, 1995),
these extensions are not very practical for large systems (beyond five or six vari-
ables). The only approaches within the GARCH framework which may be extended
to substantial systems are the Orthogonal GARCH model and the Dynamic Condi-
tional Correlation model of Engle (2002), and even these are not easily applied in
the context of forecasting a large system of equations. The GARCH structure also
imposes a parametric form on the way the density of the forecast variable evolves
which may not always be reasonable.
As a result of these limitations, a number of studies have emerged recently which
bring together two strands of the literature which both allow fairly general time
variation in forecasting models. The first of these strands introduces VAR mod-
els which allow time variation in the coefficients; this literature includes Canova
(1993), Sims (1993), Stock and Watson (1996) and Cogley and Sargent (2001). The
second strand allows for stochastic volatility in the error process of multivariate
systems; this includes work by Harvey, Ruiz and Shephard (1994), Jacquier, Polson
and Rossi (1995), Kim, Shephard and Chib (1998) and Chib, Nardari and Shephard
(2006). Allowing both time variation in the parameters and an error term with
stochastic volatility potentially allows considerable variation in the density fore-
cast. Macroeconomists have found this helpful when seeking to explain the “Great
Moderation,” namely the apparent decline in the volatility of both inflation and
output growth in the US since the mid 1980s (see Blanchard and Simon, 2001;
Stock and Watson, 2002).
A good example of this approach to generating density forecasts from a complex
model is Cogley, Morosov and Sargent (2005), who develop a forecasting Bayesian