Bayesian Inference for Time series with Infinite Variance Stable Innovations

Abstract

This article describes the use of sampling based Bayesian inference for infinite variance stable distributions and for time series with infinite variance stable innovations. For time series, an advantage of the Bayesian approach is that it enables the simultaneous estimation of the parameters characterizing the stable law, together with the parameters of the univariate or multivariate linear ARMA model. Our approach uses a Metropolis- Hastings algorithm to generate samples from the joint posterior distribution of all the parameters and is an extension to univariate and multivariate time series processes of the approach in [Bu] for independent observations. 1. Introduction A random variable X has a stable distribution S(ff; fi; ffi; oe) if there are parameters 0 ! ff 2, \Gamma1 fi 1, oe ? 0 and \Gamma1 ! ffi ! 1 such that its characteristic function has the form ([GK]): E(e itx ) = ae exp(\Gammajoetj ff (1 \Gamma ifisign(t) tan(ßff=2) + iffit) if ff 6= 1 exp(\Gammajoetj(1 + 2if..