Nonparametric estimation of volatility models with serially dependent innovations

We are interested in modelling the time series process yt=σ(xt)εtyt=σ(xt)εt, where εt=φ0εt-1+vtεt=φ0εt-1+vt. This model is of interest as it provides a plausible linkage between risk and expected return of financial assets. Further, the model can serve as a vehicle for testing the martingale difference sequence hypothesis, which is typically uncritically adopted in financial time series models. When xtxt has a fixed design, we provide a novel nonparametric estimator of the variance function based on the difference approach and establish its limiting properties. When xtxt is strictly stationary on a strongly mixing base (hereby allowing for ARCH effects) the nonparametric variance function estimator by Fan and Yao [1998. Efficient estimation of conditional variance functions in stochastic regression. Biometrika 85, 645–660] can be applied and seems very promising. We propose a semiparametric estimator of φ0φ0 that is T-consistent, adaptive, and asymptotic normally distributed under very general conditions on xtxt.