This thesis is focused on presenting novel filtering methodologies for modelling and analysing features of financial and environmental time series. The methodology used in the chapters is based on the novel observation-driven dynamic conditional score (DCS) class of time series models.

The first chapter sets up a DCS model based on the Generalised Beta of the second kind conditional distribution for modelling realized volatility (RV). Given its general formulation, it is capable of effectively capturing the long memory of the RV process with two first-order components, together with a day of the week effect both levels and in logarithms. It is also able to model and detect heteroscedasticity in log RV, which may be interpreted as the dynamic tail index in levels. Its forecasting performance is compared with that of fractional integrated models and the popular heterogeneous autoregressive model.

The second chapter examines the occurrence of extreme observations over time. It introduces a dynamic DCS model for the shape parameter that governs the tail index based on the Generalised t family of conditional distributions. The model also allows for asymmetry with different dynamics for the left and right tail index parameters. In addition the chapter introduces and studies the size and power properties of a new Lagrange multiplier test to detect the presence of dynamics in the tail index parameter.

The third chapter provides a novel methodology based on a cylindrical distribution for the joint modelling of circular and linear variables such as wind direction and velocity. The model is capable of exploiting additional information from the data to provide a new simple solution to model both data series even in absence of wind. The chapter provides an extension which allows to test and model the variations over time of the concentration parameter, which can be interpreted as time varying volatility of wind direction. The framework is further extended to allow for the possibility of modelling multiple regimes in wind direction giving dynamics directly to the switching probabilities. This approach is shown to provide significantly superior fit in comparison with the standard hidden Markov model framework used in the literature.