Essays in Robust and Nonlinear Time Series Models.

This PhD dissertation deals with the world of multivariate time series models where the behaviour of the observed process is described by using a time-varying parameter. In particular, this thesis explore three different dynamic multivariate nonlinear models which are able to deal with multivariate time series gathered from heavy-tailed phenomena. Although the popularity of linear and univariate time series models, empirical evidences have shown that variables generated from complex phenomena are typically inter-related both contemporaneously and across time. This is the case for several fields of science such as economics, finance, biology or physics, where it is widely accepted that with a univariate approach it is difficult to obtain a satisfactory representation of the reality or to make good predictions about the future. For these reasons, the literature of linear multivariate Gaussian time series models has received increasing attention. However, these models are known for their unsatisfactory performances when the collected data are contaminated by outliers, yielding biased estimates and unreliable forecasts. In fact, when departure from the hypothesis of normality is confirmed by the observed data, it is reasonable to switch into the realm of nonlinear or non-Gaussian time series models. Unfortunately, despite the development of recent technologies, the estimation of nonlinear time series models might be really challenging, since they require simulation-based and computer-intensive methods. In addition, statistical properties of such estimators are not always easy to be derived. This thesis contributes to the literature by defining dynamic multivariate and heavy-tailed models that are relatively simple. The emphasis is models which are analytically tractable and can be easily estimated by means of maximum likelihood. For each of the models, a very detailed statistical and asymptotic analysis it is provided. Their practical usefulness is highlighted with several simulation studies and empirical applications.