Wavelet Estimation For Nonparametric VarianceFunction Model Under Dependent Innovations

Volatility clustering and heteroskedasticity always exist in financial time series such as the bonds, exchange rates, stock prices and financial futures. A model, which can reflect this phenomenon, is the nonparametric variance function model. Problems on variance function in the model have attract much attention. Especially, estimation for the variance function has been a hot issue in some fields such as econometrics, financial time series and nonparametric statistics.Wavelet method, a new tool for statistical analysis of data in nearly years, has some good properties, e.g., lower smoothness requirements for nonparametric function and superior frequency localization properties. We consider a nonparametric variance model yi=σ(xi)εi,i=1,2,…,n, where xi are fixed design points, σ2(x) is an unknown nonparametric variance function and{εi} is a strictly stationary and α-mixing innovation process. The variance function model yi=σ(xi)εi is transformed into the form of nonparametric regression function yi2=σ2(xi)+σ2(xi(εi2-1), which is similar to the ordinary nonparametric regression model y=f{x)+e. Based on the wavelet estimation method of nonparametric regression function, wavelet estimate of variance function is constructed similarly. Some mild assumptions are given in the article. Some large sample properties are obtained and the application example of wavelet estimation method is given.In chapter1, we review the importance, significance and research trends for nonparametric variance models, summarize the applications of the variance function to the financial markets and economic, and give the main contents of this thesis.In chapter2, wavelet estimate of the variance function is constructed, some sample properties are discussed under certain assumptions. These properties include consistency, asymptotic normality, uniform convergence rates, etc.In chapter3, an application example of actual financial data (CNY/USD foreign exchange rate) is given.In chapter4, some conclusions are given, and future problems are presented.