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

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This report discusses the background of trigonometric sequences and series related to defining the sine and cosine functions. Proofs involving converging trigonometric sequences and series are presented using nontraditional methods. To conclude, an application of trigonometric sequences and series is shown. / text

published_or_final_version / Mathematics / Master / Master of Philosophy

published_or_final_version / Statistics / Doctoral / Doctor of Philosophy

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The exponential model of Bloomfield (1973) is becoming increasingly important due to its recent applications to long memory time series. However, this model has received little consideration in the context of short memory time series. Furthermore, there has been very little attempt at using the EXP model as a model to analyze observed time series data. This dissertation research is largely focused on developing new methods to improve the utility and robustness of the EXP model. Specifically, a new nonparametric method of parameter estimation is developed using wavelets. The advantage of this method is that, for many spectra, the resulting parameter estimates are less susceptible to biases associated with methods of parameter estimation based directly on the raw periodogram. Additionally, several methods are developed for the validation of spectral models. These methods test the hypothesis that the estimated model provides a whitening transformation of the spectrum; this is equivalent to the time domain notion of producing a model whose residuals behave like the residuals of white noise. The results of simulation and real data analysis are presented to illustrate these methods.

No description available.