A Metropolis-Hastings algorithm for reduced rank covariance matrices with application to Bayesian factor models

Markov Chain Monte Carlo (MCMC) algorithms for Bayesian factor models are generally parametrized in terms of the loading matrix and the latent common factors which are sampled into two separate blocks. In this paper, we propose a novel implementation of the MCMC algorithm that is designed for the model parametrized in terms of the reduced rank covariance matrix underlying the factor model. Hence, the strategy proposed makes it possible to sample directly from the singular covariance matrix. For this purpose, we develop an efficient Metropolis-Hastings (M-H) step that takes explicitly into account the curved geometry of the support of the target distribution. The developed M-H proposal distribution is based on a mixture of Wishart Singular distributions. The M-H step is general and can be applied, in principle, to any Bayesian model in which inference on a random positive semi-definite matrix is required. Moreover, we propose a Bayesian point estimator which preserves the original structure of the singular covariance matrix. In the paper we implement the MCMC algorithm for the static factor model and the multivariate local level model with common trends and present two empirical illustrations on financial data. The algorithm works very well in both applications.