Separation-based parameterization strategies for estimation of restricted covariance matrices in multivariate model systems

Abstract

Many multivariate model systems involve the estimation of a covariance matrix that must be positive-definite. A common strategy to ensure positive definiteness of the covariance matrix is through the use of a Cholesky parameterization of the covariance matrix. However, several model systems require imposing restrictions on the elements of the covariance elements. For instance, modelling systems may require fixing some (or all) of the diagonal elements in the covariance matrix to unity due to identification considerations. However, imposing such restrictions using the traditional Cholesky decomposition approach is not feasible and requires the additional parameterization of the Cholesky elements.

In this paper, we explore a separation-based strategy with spherical parameterization of the Cholesky matrix to impose restrictions on the covariance matrix. Importantly, using this separation-based parameterization strategy, we also explore the possibility of restricting some covariance (or correlation) terms to zero. The effectiveness of the proposed strategy is assessed through extensive simulation experiments. The results from the simulation experiments highlight better performance of the separation-based strategy in terms of recovery of model parameters – particularly those in the covariance matrix, than the traditional Cholesky parameterization approach. Finally, the proposed strategy is implemented in a joint multivariate binary probit ordered probit model system to analyze the usage (and the extent of use) of non-private modes of transportation in Bengaluru, India. In doing so, the proposed strategy is implemented to restrict several correlations to zero, thus avoiding the estimation of a profligate correlation matrix and substantially easing the estimation process.