On some new threshold-type time series models

Abstract

The subject of time series analysis has drawn significant attentions in recent years, since it is of tremendous interest to practitioners, as well as to academic researchers, to make statistical inferences and forecasts of future values of the interested variables. To do forecasting, parametric models are often required to describe the patterns of the observed data set. In order to describe data adequately, such statistical models should be established based on fundamental principles.

Two threshold-type time series models, the buffered threshold autoregressive (BAR) model and the threshold moving-average (TMA) model are studied in this thesis.

The most important contribution of this thesis is the extension of the classical threshold models via regime switching mechanisms that exhibit hysteresis to a new model called the buffered threshold model. For this type of new models, there is a buffer zone for the regime switching mechanism. The self-exciting buffered threshold autoregressive model has been thoroughly studied: a sufficient condition is given for the geometric ergodicity of the two-regime BAR process; the conditional least squares estimation is considered for the parameter estimation of the BAR model, and asymptotic properties including strong consistency and asymptotic distributions of the least square estimators are also derived. Monte Carlo experiments are conducted to give further support to the methodology developed for the new model. Two empirical examples are used to demonstrate the importance of the BAR model. Potential extensions for the basic buffer processes are discussed as well. Such extensions are expected to follow the development of classical threshold model and are motivated by their relationships with phenomena in the physical sciences.

The proposed buffer process is more general than the classical threshold model, and it should be able to capture more nonlinear features exhibited by this nonlinear world than its predecessor. Although the theoretical understanding of the model is still at its infancy, it is believed that the buffer process will provide both researchers and practitioners with a useful tool to understand the nonlinear world.

Moreover, some statistical properties of the threshold moving-average models are studied. Computer simulations have been extensively used, and some mathematical interpretation is attempted in the light of some existing research works. The model-building procedure for the TMA models is also reviewed. The effectiveness of some classical information criteria in selecting the correct TMA model is studied. A goodness-of-fit test is derived which would be useful in diagnostic checking the fitted TMA models.