Generalized Pareto models with time-varying tail behavior ∗

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. A two component mixture model is used for splitting the data into the extreme regime and the central part. Extremes beyond a threshold are assumed to follow a generalized Pareto distribution (GPD) and the parameters of the GPD are allowed to vary stochastically with time, thus inducing temporal dependence. Temporal variation and dependence is introduced at a latent level via the use of dynamic linear models (DLM). 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: the returns of Petrobras stocks and Bovespa index, all of which exhibit several extreme events. Results show advantage of our proposal over currently entertained models such as stochastic volatility, with improved estimation of high quantiles and extremes.