Multivariate Stochastic Variance Models

Changes in variance, or volatility, over time can be modelled using the approach based on autoregressive conditional heteroscedasticity (ARCH). However, the generalizations to multivariate series can be difficult to estimate and interpret. Another approach is to model variance as an unobserved stochastic process. Although it is not easy to obtain the exact likelihood function for such stochastic variance models, they tie in closely with developments in finance theory and have certain statistical attractions. This article sets up a multivariate model, discusses its statistical treatment and shows how it can be modified to capture common movements in volatility in a very natural way. The model is then fitted to daily observations on exchange rates.