On the Relationship between Uhlig Extended and beta‐Bartlett Processes

Abstract

Stochastic volatility processes are used in multi‐variate time series analysis to track time‐varying patterns in covariance matrices. Uhlig extended (UE) and beta‐Bartlett (BB) processes are especially convenient for analyzing high‐dimensional time series because they are conjugate with Wishart likelihoods. In this article, we show that UE and BB are closely related, but not equivalent: their hyperparameters can be matched so that they have the same forward‐filtered posteriors and one‐step ahead forecasts, but different joint (smoothed) posterior distributions. Under this circumstance, Bayes factors cannot discriminate the models and alternative approaches to model comparison are needed. We illustrate these issues in a retrospective analysis of volatilities of returns of foreign exchange rates. Additionally, we provide a backward sampling algorithm for the BB process, for which retrospective analysis had not been developed.