A special Cholesky-based parameterization for estimation of restricted correlation matrices

Abstract

Estimating a valid correlation matrix with structural restrictions presents significant challenges, particularly in ensuring positive definiteness and enforcing zero-correlation constraints. Traditional approaches, such as the Cholesky decomposition, often suffer from numerical instability and convergence failures in these settings. This paper introduces a novel Cholesky-based parameterization that effectively addresses these issues by allowing zero constraints while maintaining positive definiteness and unit diagonal elements. Through extensive Monte Carlo simulations, we demonstrate that the proposed method outperforms the existing spherical parameterization approach, achieving superior convergence rates, enhanced estimation accuracy, and robustness under high-correlation scenarios. An empirical application on non-commuters’ activity participation in Shanghai further validates the practical effectiveness of the proposed method, showcasing its ability to capture complex behavioral relationships while ensuring stable estimation. The results suggest that the proposed parameterization provides a reliable and computationally efficient alternative for correlation matrix estimation in multivariate models.