Some parametric and semiparametric models for financial time series analysis

Abstract

Motivated by Ling's (2007) DAR (p) model, in this thesis, we study new classes of GARCH and GARCH-in-mean models which have applications to financial data such as treasury bill rate and stock indices. Unlike the previous models in the literature, the conditional variances in our considered models are specified as functions of the time-lagged observable returns instead of the usual unobservable errors. Such a setting for the conditional variance enables us to give some new insights in the analysis of financial time series. Under the framework of an alternative specification in the conditional variance, this study considers the following aspects. First, we generalize Ling's (2007) DAR (p) model by considering a piecewise linear conditional mean instead of the single linear conditional mean in the existing models. Issues about parameter estimation and threshold test are discussed. Secondly, for a specific parametric GARCH-M model, we study its ergodicity conditions. Under some regularity assumptions, it can be shown that the quasi maximum likelihood estimator for the model is asymptotically normal. We then attempt to investigate the relationship between risk (conditional variance) and return (conditional mean) based on a class of semiparametric GARCH-M models, in which the conditional mean is specified as an unknown smooth function and the conditional variance is set as a known parametric function of lagged returns. Approaches are given to estimate the unknown function and parameters. Moreover, motivated by the time varying property of the risk aversion and the functional coefficient autoregressive model, we propose a functional coefficient autoregressive GARCH-M model to capture the variation of the risk aversion. By treating the risk aversion as a function of one day lagged return, we are able to study how yesterday's return affects today's risk magnitude. Estimates for the unknown function and parameters are discussed. Finally, we generalize the proposed functional coefficient autoregressive GARCH-M model to functional coefficient GARCH-M model, from which, we can describe the effect of common factors to risk aversion. Improved estimators for the parameters are given and, under some regularity conditions, we can prove that the parametric estimators are consistent. For all the proposed models, simulations are conducted to assess the performance of the related approaches. Applications to real data are also considered. It is demonstrated that our studied models can have comparable or better fitting performance as compared to other well known models.