The generalized ARCH or GARCH model (Bollerslev; 1986) is quite popular as a basis for analyzing the risk of financial investments. Examples are the estimation of value-at-risk (VaR) or the expected shortfall from a time series of log returns. In practice, a GARCH process of order (1,1) often provides a reasonable description of the data. In the following, we restrict ourselves to that case.
We call
a (strong) GARCH (1,1) process if
In spite of its popularity, the GARCH model has one drawback: Its symmetric
dependence on past returns does not allow for including the leverage effect
into the model, i.e. the frequently made observation that large negative
returns of stock prices have a greater impact on volatility than large
positive returns. Therefore, various parametric modifications like the
exponential GARCH (EGARCH) or the threshold GARCH (TGARCH) model
have been
proposed to account for possible asymmetric dependence of
volatility on returns. The TGARCH model, for example, introduces an additional
term into the volatility equation allowing for an increased effect of negative
on
:
We remark that the volatility function cannot be estimated by common
kernel or local polynomial smoothers as the volatilities
are not
observed directly.
Bühlmann and McNeil (1999) have considered an iterative algorithm. First, they fit
a common parametric GARCH(1,1) model to the data from which they get sample
volatilities
to replace
the unobservable true volatilities. Then, they use a common bivariate
kernel estimate to estimate
from
and
.
Using this preliminary estimate for
they obtain new sample volatilities
which are used for a further kernel estimate of
.
This procedure is iterated several times until the estimate stabilizes.
Alternatively, one could try to fit a nonparametric ARCH model of high order to the data
to get some first approximations
to
and
then use a local linear estimate based on the approximate relation