Particle Learning for Fat-Tailed Distributions

It is well known that parameter estimates and forecasts are sensitive to assumptions about the tail behavior of the error distribution. In this article, we develop an approach to sequential inference that also simultaneously estimates the tail of the accompanying error distribution. Our simulation-based approach models errors with a t_ν-distribution and, as new data arrives, we sequentially compute the marginal posterior distribution of the tail thickness. Our method naturally incorporates fat-tailed error distributions and can be extended to other data features such as stochastic volatility. We show that the sequential Bayes factor provides an optimal test of fat-tails versus normality. We provide an empirical and theoretical analysis of the rate of learning of tail thickness under a default Jeffreys prior. We illustrate our sequential methodology on the British pound/U.S. dollar daily exchange rate data and on data from the 2008–2009 credit crisis using daily S&P500 returns. Our method naturally extends to multivariate and dynamic panel data.