On the degrees of freedom in MCMC-based Wishart models for time series data

The Wishart distribution has long been a useful tool for modeling covariance structures. According to Gyndikin’s theorem, the degrees of freedom (df) for a Wishart distribution can be any real number belonging to the Gyndikin set, either integer-valued or fractional. However, the fractional-df versioned Wishart distribution has received only limited attention, which may lead to inaccurate implementation in practice. This paper shows by a numerical example that, when implementing Markov chain Monte Carlo (MCMC) methods in Wishart models for time series data, the lack of attention to the fractional df where necessary can result in seriously biased posterior estimation due to the compounding errors caused by the time dependency assumption. We further conduct a sensitivity analysis to explain why the seemingly small difference between the integer-valued df and the fractional df leads to very different outcomes.