Approximate Factor Models with a Common Multiplicative Factor for Stochastic Volatility

Common factor stochastic volatility (CSV) models capture the commonality that is often observed in volatility patterns. However, they assume that all the time variation in volatility is driven by a single multiplicative factor. This paper has two contributions. Firstly we develop a novel CSV model in which the volatility follows an inverse gamma process (CSV-IG), which implies fat Student’s t tails for the observed data. We obtain an analytic expression for the likelihood of this CSV model, which facilitates the numerical calculation of the marginal and predictive likelihood for model comparison. We also show that it is possible to simulate exactly from the posterior distribution of the volatilities using mixtures of gammas. Secondly, we generalize this CSV-IG model by parsimoniously substituting conditionally homoscedastic shocks with heteroscedastic factors which interact multiplicatively with the common factor in an approximate factor model (CSV-IG-AF). In empirical applications we compare these models to other multivariate stochastic volatility models, including different types of CSV models and exact factor stochastic volatility (FSV) models. The models are estimated using daily exchange rate returns of 8 currencies. A second application estimates the models using 20 macroeconomic variables for each of four countries: US, UK, Japan and Brazil. The comparison method is based on the predictive likelihood. In the application to exchange rate data we find strong evidence of CSV and that the best model is the IG-CSV-AF. In the Macro application we find that 1) the CSV-IG model performs better than all other CSV models, 2) the CSV-IG-AF is the best model for the US, 3) the CSV-IG is the best model for Brazil and 4) exact factor SV models are the best for UK and JP.