With the analysis of (financial) time series one of the most important goals is to produce forecasts. Using past observed data one would like to make some statements about the future mean, the future volatility, etc., i.e., one would like to estimate the expectation and variance of the underlying process conditional on the past. One method to produce such estimates will be introduced in this chapter.
Let
be a time series. We consider a
nonlinear autoregressive heteroscedastic model of the form
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With the specific choice
and
the process
is an AR(1) process. Every strong
ARCH(1) process
satisfies the model (13.1). In
this case
, and it holds that
with the parameters
and
, compare
Definition 12.1. With respect to the structure of the
conditional mean and the conditional variance, the model above is
another broad generalization of the (strong) ARCH models. The
advantage of this nonparametric Ansatz is that the model
contains no structural assumptions about the functions
and
, since such assumptions are often not supported by
observations in the data.
Closely related to our model is the Qualitative Threshold ARCH
model (QTARCH) studied in Gouriéroux and Monfort (1992), which for the
case of one lag (QTARCH(1)) is a special case of
(13.1), where the unknown functions
and
take the form of step functions - see (12.31). On the
other hand (13.1) can also be described under certain
regularity assumptions on
and
as a limit model of the
QTARCH(1) models when
, in that
and
are approximated with elementary functions.
The work of Gouriéroux and Monfort is the first to consider the
conditional mean and the conditional variance together in a
nonparametric model. The applications of this idea introduced here
are taken from Härdle and Tsybakov (1997), and are also considered
independently in Franke et al. (2002). In the following we will
construct a class of estimators based on the local polynomial
regression for the conditional volatility
and the
conditional mean
of the time series
under the model
assumptions (13.1).
In addition to the model assumptions (13.1) certain
regularity assumptions, although no structural assumptions, on
and
will be made. As the main result of this chapter we will
show that this combined estimation of the conditional expectation
and the conditional volatility is asymptotically normally
distributed.