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