The model (13.1) is thoroughly studied together with
financial time series, in particular under the assumptions of the
ARCH structure, in Engle (1982). Until recently academic
research focused mainly on the (linear) conditional mean, or it
was assumed that the conditional variance was constant or, as in
the ARCH models, that it had a special form. At the beginning of
the eighties this deficit in the literature was corrected by
Engle (1982),
and Robinson (1984,1983) and in the
statistic literature by Collomb (1984)
and Vieu (1995). There have also been nonparametric and
semi-parametric approximations suggested in Gregory (1989),
Engle and Gonzalez-Rivera (1991). Since then the interest in the
nonparametric situation discussed here, in which the form of the
functions and
is not identified ahead of time, has clearly
grown in the economics and statistics literature, see
Fan and Yao (2003).
The QTARCH models (12.31) in Gouriéroux and Monfort (1992)
create a generalization of the threshold models for the
conditional mean in Tong (1983). The methods
from Gouriéroux and Monfort (1992) and the article from
McKeague and Zhang (1994) are based on histogram estimations of the
volatility. The works from
Chen and Tsay (1993b,a)
concentrate on additive modelling of the the mean function .
Additive or multiplicative structures of volatility are considered
in Härdle et al. (1997), Yang et al. (1999) and
Hafner (1998). The general nonparametric ARCH model is
handled in Härdle et al. (1996). Franke (1999)
discusses the connection between the nonparametric AR-ARCH model
and the discrete version of geometric Brownian motion which is
used as a foundation for the Black-Scholes applications.
Franke, Härdle and Kreiss (2003) study in connection to a special stochastic
volatility model a nonparametric de-convolution estimator for the
volatility function as the first step towards the nonparametric
handling of general GARCH models.
The idea of the local polynomial estimation originates in Stone (1977), Cleveland (1979) and Katkovnik (1985,1979), who have used it on nonparametric regression models. Statistical properties of LP estimators by nonparametric regression models (convergence, convergence rate and pointwise asymptotic normality) are derived in Tsybakov (1986). References to more recent studies in this area can be found in Fan and Gijbels (1996).
Apart from the statistical studies of the model (13.1),
the utilized theoretical probability properties of the constructed
process are also of importance. This is studied in the
works of Doukhan and Ghindès (1981),
Chan and Tong (1985), Mokkadem (1987),
Diebolt and Guégan (1990) and Ango Nze (1992). In these
articles the ergodicity, geometric ergodicity and mixture
properties of the process
are derived.