On the observed-data deviance information criterion for volatility modeling

We propose importance sampling algorithms based on fast band matrix routines for estimating the observed-data likelihoods for a variety of stochastic volatility models. This is motivated by the problem of computing the deviance information criterion (DIC)-a popular Bayesian model comparison criterion that comes in a few variants. Although the DIC based on the conditional likelihood-obtained by conditioning on the latent variables-is widely used for comparing stochastic volatility models, recent studies have argued against its use on both theoretical and practical grounds. Indeed, we show via a Monte-Carlo study that the conditional DIC tends to favor overfitted models, whereas the DIC based on the observed-data likelihood-calculated using the proposed importance sampling algorithms-seems to perform well. We demonstrate the methodology with an application involving daily returns on the Standard & Poors 500 index.