Temporal dependence in extremes with dynamic model

Abstract

This paper is concerned with the analysis of time series data with temporal dependence through extreme events. This is achieved via a model formulation that considers separately the central part and the tail of the distributions, using a two component mixture model. Extremes beyond a thresh old are assumed to follow a generalized Pareto distribution (GPD). Temporal dependence is induced by allowing to GPD parameter to vary with time. Temporal variation and dependence is introduced at a latent level via the novel use of dynamic linear models (DLM). Novelty lies in the time variation of the shape and scale parameter of the resulting distribution. These changes in limiting regimes as time changes reflect better the data behavior, with importante gains in estimation and interpretation. The central part follows a nonparametric mixture approach. The uncertainty about the threshold is explicitly considered. Posterior inference is performed through Markov Chain Monte Carlo (MCMC) methods. A variety of scenarios can be entertained and include the possibility of alternation of presence and absence of a finite upper limit of the distribution for different time periods. Simulations are carried out in order to analyze the performance of our proposed model. We also apply the proposed model to financial time series: returns of Petrobr´as stocks and SP500 index. Results show advantage of our proposal over currently entertained models such as stochastic volatility, with improved estimation of high quantiles and extremes.