The Spherical Parametrisation for Correlation Matrices and its Computational Advantages

In this paper, we analyse the computational advantages of the spherical parametrisation for correlation matrices in the context of Maximum Likelihood estimation via numerical optimisation. By using the special structure of correlation matrices, it is possible to define a bijective transformation of an $$n \times n$$ n × n correlation matrix $$R$$ R into a vector of $$n(n-1)/2$$ n ( n - 1 ) / 2 angles between 0 and $$\pi$$ π . After discussing the algebraic aspects of the problem, we provide examples of the use of the technique we propose in popular econometric models: the multivariate DCC-GARCH model, widely used in applied finance for large-scale problems, and the multivariate probit model, for which the computation of the likelihood is typically accomplished by simulated Maximum Likelihood. Our analysis reveals the conditions when the spherical parametrisation is advantageous; numerical optimisation algorithms are often more robust and efficient, especially when R is large and near-singular.