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.