226 The Basics of financial economeTrics
What Do ARCH/GARCH Models Represent?
Thus far we have described ARCH and GARCH models as models of returns
or of financial time series. In these models, volatility coincides with the mag-
nitude of returns or of some other financial time series. This is because we
assume that the series to be modeled is proportional to a zero-mean i.i.d.
sequence as given by equation (11.5).
However, in many applications this is not the case. In fact, ARCH/
GARCH models might describe the behavior of the errors of some more
fundamental model of the conditional mean. In this case, volatility rep-
resents the magnitude of unpredictable residuals. In other words, we
first model the conditional mean and then we model residuals as ARCH/
GARCH models. Note that because we are dealing with weakly stationary
processes, both the unconditional mean and the unconditional volatility
are constants.
For example, we might represent the returns of stocks with some fac-
tor model where returns are regressed over a number of predictors. In this
case, forecasts of returns conditional to the predictors are obtained through
the primary regressions over the factors; volatility is the magnitude of fluc-
tuations of the residuals of the factor model. The ARCH/GARCH model
applies to the residuals from previous modeling of the conditional mean.
The model would be written as follows:
Rftt=+αβ+ut (11.11)
udtt=+σεt (11.12)
σσ^2 tt=+ca 11 ua^22 −−++ ptubpt++ 1122 −−+bqtσ q (11.13)
In this model, returns are determined by equation (11.11) in function
of factors. A GARCH model then describes the residuals ut. That is, we first
write a model of conditional mean as a regression of returns on a number
of factors, given by the equation (11.11). The residuals of this model, which
represent the unpredictable component, do not have a constant variance but
are subject to GARCH modeling.
Univariate Extensions of GARCH Modeling
The principle of GARCH modeling consists of making residuals (or eco-
nomic variables) proportional to a volatility process that is modeled
separately. Subsequent to the initial introduction of GARCH models, many