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. / published_or_final_version / Statistics and Actuarial Science / Doctoral / Doctor of Philosophy

With the increasingly economic activities, people have more and more interest in econometric models. There are two mainstream econometric models which are very popular in recent decades. One is quantile autoregressive (QAR) model which allows varying-coefficients in linear time series and greatly promotes the ranges of regression research. The first topic of this thesis is to focus on the modeling of QAR model. We propose two important measures, quantile correlation (QCOR) and quantile partial correlation (QPCOR). We then apply them to QAR models, and introduce two valuable quantities, the quantile autocorrelation function (QACF) and the quantile partial autocorrelation function (QPACF). This allows us to extend the Box-Jenkins three-stage procedure (model identification, model parameter estimation, and model diagnostic checking) from classical autoregressive models to quantile autoregressive models. Specifically, the QPACF of an observed time series can be employed to identify the autoregressive order, while the QACF of residuals obtained from the model can be used to assess the model adequacy. We not only demonstrate the asymptotic properties of QCOR, QPCOR, QACF and PQACF, but also show the large sample results of the QAR estimates and the quantile version of the Ljung- Box test. Moreover, we obtain the bootstrap approximations to the distributions of parameter estimators and proposed measures. Simulation studies indicate that the proposed methods perform well in finite samples, and an empirical example is presented to illustrate the usefulness of QAR model. The other important econometric model is autoregressive conditional duration (ACD) model which is developed with the purpose of depicting ultra high frequency (UHF) financial time series data. The second topic of this thesis is designed to incorporate ACD model with one of the extreme value distributions, i.e. Fréchet distribution. We apply the maximum likelihood estimation (MLE) to Fréchet ACD models and derive its generalized residuals for model adequacy checking. It is noteworthy that simulations show a relative greater sensitiveness in the linear parameters to sampling errors. This phenomenon successfully reflects the skewness of the Fréchet distribution and suggests a method to practitioners in proceeding model accuracy. Furthermore, we present the empirical sizes and powers for Box-Pierce, Ljung-Box and modified Box-Pierce statistics as comparisons of the proposed portmanteau statistic. In addition to the Fréchet ACD, we also systematically analyze theWeibull ACD, where the Weibull distribution is the other nonnegative extreme value distribution. The last topic of the thesis explains the estimation and diagnostic checking the Weibull ACD model. By investigating the MLE in this model, there exhibits a slight sensitiveness in linear parameters. However, there is an obvious phenomenon on the trade-off between the skewness of Weibull distribution and the sampling error when the simulations are conducted. Moreover, the asymptotic properties are also studied for the generalized residuals and a goodness-of-fit test is employed to obtain a portmanteau statistic. Through the simulation results in size and power, it shows that Weibull ACD is superior to Fréchet ACD in specifying the wrong model. This is meaningful in practice. / published_or_final_version / Statistics and Actuarial Science / Doctoral / Doctor of Philosophy

The traditional threshold time series model is famous for its capability in capturing asymmetry. Regime switching takes place immediately when a certain variable crosses the threshold. However, this type of model may not be suitable for data which have no clear cut between regimes. A new generation of threshold type model, buffered time series model, is modified from the traditional threshold time series model. A buffer zone is introduced to replace the role of the threshold; regime switching will not take place within the buffer zone. The regime switching mechanism mimicks a climatological example and the buffered model may be suitable for data in which there is a region where the probabilistic structure of the data is insensitive to changes. Self-exciting buffered generalized autoregressive conditional heteroscedasticity (buffered GARCH) model is considered. Quasi-maximum likelihood is employed for parameter estimation. Strong consistency and asymptotic distributions are derived. Simulation experiments are carried out to verify the properties of the estimators. The buffered GARCH model is applied to two currency exchange rate data sets, US dollar to Moroccan dirham exchange rate and US dollar to Israeli new shekel exchange rate. At the same time, threshold GARCH model is also applied to the data sets in order to have comparison between the buffered GARCH model and threshold GARCH model. It is found that the buffered GARCH model beats the threshold GARCH model in terms of one information criterion, revealing that the buffered GARCH model may have advantage over the threshold GARCH model. / published_or_final_version / Statistics and Actuarial Science / Master / Master of Philosophy

Conditional heteroscedastic models are one important type of time series models which have been widely investigated and brought out continuously by scholars in time series analysis. Those models play an important role in depicting the characteristics of the real world phenomenon, e.g. the behaviour of _nancial market. This thesis proposes a mixture double autoregressive model by adopting the exibility of mixture models to the double autoregressive model, a novel conditional heteroscedastic model recently proposed by Ling (2004). Probabilistic properties including strict stationarity and higher order moments are derived for this new model and, to make it more exible, a logistic mixture double autoregressive model is further introduced to take into account the time varying mixing proportions. Inference tools including the maximum likelihood estimation, an EM algorithm for searching the estimator and an information criterion for model selection are carefully studied for the logistic mixture double autoregressive model. We notice that the shape changing characteristics of the multimodal conditional distributions is an important feature of this new type of model. The conditional heteroscedasticity of time series is also well depicted. Monte Carlo experiments give further support to these two new models, and the analysis of an empirical example based on our new models as well as other mainstream ones is also reported. / published_or_final_version / Statistics and Actuarial Science / Master / Master of Philosophy

published_or_final_version / Statistics / Doctoral / Doctor of Philosophy

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Thesis (Ph. D.)--University of Newcastle, 2000. / Department of Civil, Surveying and Environmental Engineering. Includes bibliographical references (leaves R-1-R-9). Also available online.

Thesis (M. Sc.)--School of Mathematics and Statistics, Faculty of Science, University of Sydney, 2004. / Bibliography: leaves 129-135.

Thesis (M.Sc. (Course Work) (Mathematical Statistics)) -- University of Pretoria, 2007. / Includes bibliographical references. Available on the Internet via the World Wide Web.