Title: A note on conditional covariance matrices for elliptical distributions,
Abstract: In this short note we provide an analytical formula for the conditional
covariance matrices of the elliptically distributed random vectors, when the
conditioning is based on the values of any linear combination of the marginal
random variables. We show that one could introduce the univariate invariant
depending solely on the conditioning set, which greatly simplifies the
calculations. As an application, we show that one could define uniquely defined
quantile-based sets on which conditional covariance matrices must be equal to
each other if only the vector is multivariate normal. The similar results are
obtained for conditional correlation matrices of the general elliptic case.