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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: 