Nonlinear long memory models with applications in finance
The last decade has witnessed a great deal of research in modelling volatility of financial asset returns, expressed by time-varying variances and covariances. The importance of modelling volatility lies in the dependence of any financial investment decision on the expected risk and return as formalized in classical asset pricing theory. Precise evaluation of volatilities is a compulsory step in order to perform correct options pricing according to recent theories of the term structure of interest rates and for the construction of dynamic hedge portfolios. Models of time varying volatility represent an important ground for the development of new estimation and forecasting techniques for situations not reconcilable with the Gaussian or, more generally, a linear time series framework. This is particularly true for the statistical analysis of time series with long range dependence in a nonlinear framework. The aim of this thesis is to introduce parametric nonlinear time series models with long memory, with particular emphasis on volatility models, and to provide a methodology which yields asymptotically exact inference on the parameters of the models. The importance of these results stems from: (i) rigorous asymptotics was lacking from the stochastic volatility literature; (ii) the statistical literature does not cover the analysis of the asymptotic behaviour of quadratic forms in nonlinear non-Gaussian variates that characterizes our problem.