The GARCH(1,1) is the simplest and most robust of the family of volatility
models. However, the model can be extended and modified in many ways. I will
briefly mention three modifications, although the number of volatility models that
can be found in the literature is now quite extraordinary.
The GARCH(1,1) model can be generalized to a GARCH(p,q) model—that
is, a model with additional lag terms. Such higher-order models are often useful
when a long span of data is used, like several decades of daily data or a year of
hourly data. With additional lags, such models allow both fast and slow decay of
information. A particular specification of the GARCH(2,2) by Engle and Lee
(1999), sometimes called the “component model,” is a useful starting point to this
approach.
ARCH/GARCH models thus far have ignored information on the direction of
returns; only the magnitude matters. However, there is very convincing evidence
that the direction does affect volatility. Particularly for broad-based equity indices
and bond market indices, it appears that market declines forecast higher volatility
than comparable market increases do. There is now a variety of asymmetric GARCH
models, including the EGARCH model of Nelson (1991), the TARCH model—
threshold ARCH—attributed to Rabemananjara and Zakoian (1993) and Glosten,
Jaganathan and Runkle (1993), and a collection and comparison by Engle and Ng
(1993).
The goal of volatility analysis must ultimately be to explain the causes of
volatility. While time series structure is valuable for forecasting, it does not
satisfy our need to explain volatility. The estimation strategy introduced for
ARCH/GARCH models can be directly applied if there are predetermined or
exogenous variables. Thus, we can think of the estimation problem for the
variance just as we do for the mean. We can carry out specification searches and
hypothesis tests to find the best formulation. Thus far, attempts to find the
ultimate cause of volatility are not very satisfactory. Obviously, volatility is a
response to news, which must be a surprise. However, the timing of the news
may not be a surprise and gives rise to predictable components of volatility, such
as economic announcements. It is also possible to see how the amplitude of
news events is influenced by other news events. For example, the amplitude of
return movements on the United States stock market may respond to the
volatility observed earlier in the day in Asian markets as well as to the volatility
observed in the United States on the previous day. Engle, Ito and Lin (1990) call
these “heat wave” and “meteor shower” effects.
A similar issue arises when examining several assets in the same market. Does
the volatility of one influence the volatility of another? In particular, the volatility
of an individual stock is clearly influenced by the volatility of the market as a whole.
This is a natural implication of the capital asset pricing model. It also appears that
there is time variation in idiosyncratic volatility (for example, Engle, Ng and
Rothschild, 1992).