Linear Models and Time-Series Analysis

14Multivariate Mixture Distributions

Occasionally, papers are published suggesting how returns can be forecast using a simple statistical model, and presumably these techniques are the basis of the decisions of some financial analysts. More likely the results are fragile: once you try to use them, they go away.

(Clive W. J. Granger, 2005, p. 36)

The next obvious step is towards using predictive, or conditional, distributions. Major problems remain, particularly with parametric forms and in the multivariate case. For the center of the distribution a mixture of Gaussians appears to work well but these do not represent tail probabilities in a satisfactory fashion.

(Clive W. J. Granger, 2005, p. 37)

Use of the i.i.d. univariate discrete mixture of normals distribution, or MixN, as detailed in Chapter III.5.1, allows for great enrichment in modeling flexibility compared to the Gaussian. Here, we extend this to the multivariate case. We also develop the methodology for mixtures of (multivariate) Laplace, this distribution having the same tail behavior (short, or thin tails) as the normal, but such that it is leptokurtic. This is advantageous for modeling heavier‐tailed data, such as financial asset returns. We will also see other important concepts such as mixture diagnostics and an alternative estimation paradigm for multivariate mixtures.

14.1 The Distribution

Like its univariate ...