A Balanced Parametrization of Correlation Matrices

We introduce the balanced correlation parametrization (BCP), which maps positive-definite correlation matrices one-to-one onto the set of symmetric hollow (zero-diagonal) matrices with unrestricted off-diagonal entries. A key feature of the parametrization is that both the transformation and its inverse are available in closed form. It can be viewed as a generalization of the transformation underlying the classical correlation $t$-statistic to higher dimensions. It has a direct interpretation as a scaled sum of marginal and partial correlations, is inverse-consistent and equivariant under variable permutations and sign changes, and preserves block structure. A novel property of BCP is that it preserves correlation sparsity as the dimension tends to infinity under suitable factor conditions, providing a basis for sparse high-dimensional correlation modeling. An empirical illustration using standardized idiosyncratic daily stock returns for 1,765 equities over 15 years shows that BCP retains the approximate sparsity pattern and magnitudes of shock correlations, and that it yields accurate positive-definite reconstructions under sparse coordinate restrictions.