Maximum likelihood estimation and forecasting for GARCH, Markov switching, and locally stationary wavelet processes

Financial time series are frequently met both in daily life and the scientific world. It is clearly of importance to study the financial time series, to understand the mechanism giving rise to the data, and/or predict the future values of a series. This thesis is dedicated to statistical inferences of a number of models for financial time series. Financial time series often exhibit time-varying and clustering volatility (conditional variance), which were not handled well by traditional models, until the development of the autoregressive conditionally heteroscedastic (ARCH) and the generalized ARCH (GARCH) models. We prove the consistency and asymptotic normality of the quasi-maximum likelihood estimators for a GARCH(1,2) model with dependent innovations, which extends the results for the GARCH(1,1) model in the literature under weaker conditions. The regime-switching GARCH (RS-GARCH) model extends the GARCH models by incorporating a Markov switching into the variance structure. The statistical inferences for the RS-GARCH model are difficult due to the complex dependence structure. One alternative is to take average over all regimes at every step, and adapt the integrated conditional variances. Another one is to transform the GARCH into an ARCH model. The maximum likelihood (ML) estimation of these two cases is considered. Consistency of the ML estimators is proved, and the asymptotic normality is suggested by simulation studies. The results are further generalized to a general autoregressive model with Markov switching, in which the autoregression can be of infinite order. Consistency of the ML estimators is obtained and the asymptotic normality is conjectured. Time series analysis can also be conducted in frequency domain, i.e. to analyze their spectral values obtained by e.g. Fourier or wavelet transforms. Locally stationary wavelet (LSW) processes are a class of processes defined on a set of non-decimated wavelets. We first address the problem on how to select a wavelet in practice, and some guidelines are suggested by simulation studies. The existing forecasting algorithm for LSW processes is found vulnerable to outliers, and a new forecasting algorithm is proposed to overcome this weakness. The new algorithm is shown stable and outperforms the existing algorithm when applied to real financial data. The volatility forecasting ability of LSW model based on our new algorithm is then discussed and is shown to be competitive with GARCH models. Algorithms and functions for data generation, calculation and maximization of the likelihoods for RS-GARCH models, and for the new forecasting algorithm of LSW processes are appended