Throughout my Ph.D. life, I mainly work on research topics about time series. This thesis consists of three representative works I have done for statistical modeling of time series. Modeling time series is a fundamental task in statistics, with extensive applications in finance, economics, climate etc. One significant characteristics of time series is that the data have complicated dependence structures.

For univariate time series, there is temporal dependence, where the past influences the future (autocorrelation). The modeling of univariate time series is a relatively well-studied research area, especially in financial applications. However, for the modeling of extreme events, such as maximum daily loss of a high-frequency trading, the suitable models are still rare. An accurate time series model for extreme events is essential for understanding the behavior of extreme risk and thus managing the extreme risk. Parts of the thesis ( Chapter 2) are dedicated to the development of a dynamic extreme-value-theory (EVT) based time series model which can capture the time-varying behavior of maxima in financial time series.

For multivariate time series, there are both temporal dependence of each component univariate time series and cross-sectional dependence across all the component univariate time series. To accurately capture the behavior of multivariate time series, it is essential for the time series model to be highly flexible such that it is capable of modeling various dependence structures, such as nonlinear dependence, tail dependence and asymmetric dependence. Essentially, we need to construct flexible dependence structures, i.e. multivariate distribution functions. In the literature, Copula is one of the most widely used methods for generating sophisticated multivariate distribution functions. Parts of the thesis (Chapter 3 and 4) focus on the construction of novel copulas and copula-based multivariate time series models. A complicated statistical model may be computationally expensive to estimate via the conventional maximum likelihood estimator (MLE), which can also be infeasible if the model is semiparametric. Parts of the thesis, specifically the parameter estimation sections of Chapter 2 to 4, propose efficient estimation procedures for the complex models based on stepwise estimation and composite likelihood estimation, and study the theoretical properties of irregular MLE and semiparametric sequential estimators.