Local Gaussian correlation: A new measure of dependence
It is a common view among finance analysts and econometricians that the correlation between financial objects becomes stronger as the market is going down, and that it approaches one when the market crashes, having the effect of destroying the benefit of diversification. The purpose of this paper is to introduce a local dependence measure that gives a precise mathematical description and interpretation of such phenomena. We propose a new local dependence measure, a local correlation function, based on approximating a bivariate density locally by a family of bivariate Gaussian densities using local likelihood. At each point the correlation coefficient of the approximating Gaussian distribution is taken as the local correlation. Existence, uniqueness and limit results are established. A number of properties of the local Gaussian correlation and its estimate are given, along with examples from both simulated and real data. This new method of modelling carries with it the prospect of being able to do locally for a general density what can be done globally for the Gaussian density. In a sense it extends Gaussian analysis from a linear to a non-linear environment.