We compare a number of GARCH and stochastic volatility (SV) models using nine series of oil, petroleum product and natural gas prices in a formal Bayesian model comparison exercise. The competing models include the standard models of GARCH(1,1) and SV with an AR(1) log-volatility process and more flexible models with jumps, volatility in mean and moving average innovations. We find that: (1) SV models generally compare favorably to their GARCH counterparts; (2) the jump component substantially improves the performance of the standard GARCH, but is unimportant for the SV model; (3) the volatility feedback channel seems to be superfluous; and (4) the moving average component markedly improves the fit of both GARCH and SV models. Overall, the SV model with moving average innovations is the best model for all nine series.

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. While 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 (S&P) 500 index.

The deviance information criterion (DIC) has been widely used for Bayesian model comparison. However, recent studies have cautioned against the use of certain variants of the DIC for comparing latent variable models. For example, it has been argued that the conditional DIC-based on the conditional likelihood obtained by conditioning on the latent variables-is sensitive to transformations of latent variables and distributions. Further, in a Monte Carlo study that compares various Poisson models, the conditional DIC almost always prefers an incorrect model. In contrast, the observed-data DIC-calculated using the observed-data likelihood obtained by integrating out the latent variables-seems to perform well. It is also the case that the conditional DIC based on the maximum a posteriori (MAP) estimate might not even exist, whereas the observed-data DIC does not suffer from this problem. In view of these considerations, fast algorithms for computing the observeddata DIC for a variety of high-dimensional latent variable models are developed. Through three empirical applications it is demonstrated that the observed-data DICs have much smaller numerical standard errors compared to the conditional DICs. The corresponding Matlab code is available upon request.

We compare a number of widely used trend‐cycle decompositions of output in a formal Bayesian model comparison exercise. This is motivated by the often markedly different results from these decompositions—different decompositions have broad implications for the relative importance of real versus nominal shocks in explaining variations in output. Using U.S. quarterly real GDP, we find that the overall best model is an unobserved components model with two features: (i) a nonzero correlation between trend and cycle innovations and (ii) a break in trend output growth in 2007. The annualized trend output growth decreases from about 3.4% to 1.2%–1.5% after the break. The results also indicate that real shocks are more important than nominal shocks. The slowdown in trend output growth is robust when we expand the set of models to include bivariate unobserved components models.

This paper reconciles two widely used trend-cycle decompositions of GDP that give markedly different estimates: the correlated unobserved components model yields output gaps that are small in amplitude, whereas the Hodrick-Prescott (HP) filter generates large and persistent cycles. By embedding the HP filter in an unobserved components model, we show that this difference arises due to differences in the way the stochastic trend is modeled. Moreover, the HP filter implies that the cyclical components are serially independent-an assumption that is decidedly rejected by the data. By relaxing this restrictive assumption, the augmented HP filter provides comparable model fit relative to the standard correlated unobserved components model.