7 - Multivariate Models, Correlation and Association

Summary

Dynamic conditional score models extend quite naturally to deal with changing location and scale in multivariate time series. Although a full treatment requires extensive use of matrix algebra, the main practical issues concern the nature and scope of restrictions on the parameter matrices governing the dynamics. The asymptotic theory of ML estimation is straightforward in principle, though the details are not always easy to work out.

The main challenge lies in constructing and analyzing models that embody changing relationships between variables. In this case, the asymptotic theory is no longer straightforward, even for Gaussian models. Nevertheless, even when it is not possible to develop a full asymptotic theory, the structure of the information matrix, and the conditions under which it is positive definite, provides a good indication of how well different models and parameterizations might work in practice.

There are important reasons for wishing to use multivariate models in finance. Understanding and measuring the relationship between movements in different assets play a key role in designing a portfolio. The standard theory is based on the multivariate normal distribution, and it shows how to construct a portfolio that gives minimum variance for a given expected return. Unfortunately, the multivariate normal distribution is not usually suitable for this task for two reasons: asset returns are not normally distributed, and their comovements are not adequately captured by correlation coefficients. More specifically, heavy tails are a feature of marginal distributions, and the probab-ility of two asset prices exhibiting big movements in the same direction may be much higher than it would be with a bivariate normal.