Multivariate models for dependent clusters of variables with conditional independence given aggregation variables

A general multivariate distributional approach, with conditional independence given aggregation variables, is presented to combine group-based submodels when variables are naturally divided into several non-overlapping groups. When the distributions are all multivariate Gaussian, the dependence among different groups is parsimonious based on conditional independence given linear combinations of variables in each group. For the case of multivariate t distributions in each group, a grouped t distribution is obtained. The approach can be extended so that the copula for each group is based on a skew-t distribution, and an application of this is given to financial returns of stocks in several different sectors. Another example of the modeling approach is given with variables separated into groups based on their units of measurements.