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
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
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