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### Showing that a Gaussian copula is not in general an Archimdean copula

An -dimensional Archimedean copula is one that can be represented by:

One way of showing that the Gaussian copula is not in general an Archimedean copula is to consider a three dimensional Gaussian copula. Its copula density (for a correlation matrix ) can be written as:

In general,  will have 3 different off-diagonal elements, derived from the three different correlations between  and , between  and  and between  and respectively. Thus the form of the copula density if  expressed as a function of the remaining two components of , i.e. here  and , will differ from its form if  expressed as a function of  and  etc. However, to be Archimedean, the copula needs to be indifferent between the components of .

For , the Gaussian copula has too many free parameters to be Archimedean.

Conversely, if returns are multivariate normal and have an exchangeable copula then the returns can be characterised by a factor structure involving a single factor.

A set of  random variables,  () is said to possess a factor structure if their covariance matrix, , is of the form  where  is an  matrix,  is an  matrix (and there are  factors) and  is a diagonal matrix. Suppose the variance of each  is  and we define . Then  have unit variance and their covariance (now also correlation) matrix also has the form . The copulas describing the  and  are the same. If it is exchangeable and  are multivariate normal then we must have  being the same for all , say . This arises if we set  and  as follows, if  is the identity matrix:

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