Linear Models and Time-Series Analysis

12Multivariate Distributions

The multivariate normal distribution is the right starting point for modeling many phenomena, though it will prove inadequate in several contexts. For daily stock returns, we are motivated to consider distributions that exhibit leptokurtosis in the univariate margins, as well as possible asymmetry. Distributions that nest the multivariate normal, or yield it as a limiting case, and possess desirable features enabling them to be of use in a variety of applications (in particular, but not only, empirical finance) include the multivariate generalized hyperbolic distribution, or MGHyp (Section 11.2.4), discrete mixtures of multivariate normals (Chapter 14), and the multivariate noncentral , or MVNCT (Section 12.2 below). While very flexible, the MGHyp and the MVNCT are such that each univariate margin has the same tail thickness parameter. This is not appropriate in some situations, and we present distributions that allow for heterogeneous tail behavior.

We proceed as follows. The short Section 12.1 , on the multivariate Student's distribution, serves as a warmup for Section 12.2 , on its noncentral extension. Sections 12.3 and 12.4 are dedicated to a group ...