14.4 Recommended Literature

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 $ f$ and $ s$ 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 $ f$. 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 $ (Y_{i})$ 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 $ (Y_{i})$ are derived.