I wanted to upload here a small problem I started to work on…. unfortunatley, I could not find (yet) a proper answer. Any comments and suggestions are welcomed. The problem is simple: consider 3 random variables that can be interpreted as exchange rates. Thus X is the exchange rate USD versus EUR, Y is the exchange rate EUR versus YEN, while Z is the exchange rate USD versus YEN. We then have the following constraint XY=Z. Given that constraint, what are the implied constraints on the correlation matrix of random vector (X,Y,Z) ?
- Bounds on correlation
There are classically standard constraints on correlation. For instance, from Cauchy-Schwarz’s inequality

but those bounds are not necessarily sharp, given distributions of and
. From Fréchet-Hoeffding bounds, we know that

where . Further, correlation matrices are necessarily positive-semidefinite matrices.
To go further to find possible bounds, recall that

and therefore

i.e.

Similarity, will be a function of
.
Thus, given marginal distributions of and
, covariances (and also correlations
and
) are functions of
,
and
.
Somehow, we have obtained 2 degrees of freedom: if there are no additional constraints on those two coefficients, there should be no other constraints than having positive-semidefinite matrices.
I guess that and
should not be connected. Those quantities are related to the co-skewness* and they should be different when dependence is asymmetric.
Given a 3 parameter copula for pair , respectively describing global dependence (that can be related to
and tail asymmetry (that should be related either to
and
depending on the corner) it should be possible to derive any kind of triplets.
- Running simulations
I have tried to generate a lot of samples. The idea is to draw pairs from mixtures of Marshall-Olkins’ copula (wich are asymmetric), where parameters where generated randomly. I draw 250 values according to that copula. Then, the third variable is obtained taking the product of normalized pairs (or Studentized). And then calculate the associated correlation matrix.
- draw
from
(but actually any positive distribution should work) and
- do 250 times the following (index
)
- draw
with a Bernoulli
distribution
- draw&nbs
p;from copula
if
and
if
- set
and
, and
- estimate the correlation of sample
In dimension 3, I have the following graph,




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So far I have not been able to obtain all possible correlations, I guess… If anyone has suggestions…
* Recall that the co-skewness matrix, in dimension 2, for a pair is simply

where .
OpenEdition suggests that you cite this post as follows:
Arthur Charpentier (March 18, 2010). Exchange rates and correlation matrices. Freakonometrics. Retrieved February 15, 2025 from https://doi.org/10.58079/oudh