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

Most of the proposed Markov chain Monte Carlo (MCMC) algorithms for estimating static and dynamic Bayesian factor models are 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 which 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 reduced rank covariance matrix. The alternative parameterization of the model is undoubtedly more natural for the linear dynamic factor model. Furthermore, it allows us to rewrite the static factor model as a hierarchical (multilevel) linear model. In this way, a better mixing of the Markov chain is obtained. We adopt, as prior for the singular covariance matrix, the noninformative prior distribution first considered by Diaz-Garcia and Gutierrez (2006). We implement an efficient MCMC algorithm characterized by the sampling of the singular covariance matrix and the associated (unobserved) systematic component in one block. Furthermore, we propose the sampling of the singular covariance matrix marginalized over the systematic component. For this purpose, we develop a Metropolis-Hastings (M-H) step that takes explicitly into account the curved geometry of the support of the target distribution. The proposal distribution is based on a mixture of Wishart Singular distributions; see Diaz-Garcia et al. (1997). It is worth noting that, as a result of working with singular distributions, the prior and posterior densities, as well as the density of the proposal distribution in the M-H step, are specified with respect to Hausdorff measure and integral. The curved geometry of the support has implications for the Bayesian inference on the reduced rank covariance matrix too. That is, a Bayesian point estimator which preserves the original structure of the singular covariance matrix is required. We propose a Bayesian point estimator which is obtained by the generalized Choleski decomposition for reduced rank covariance matrices. We apply our approach to static factor models and present an empirical illustration on exchange rates. Moreover, we consider a simple but important example of linear dynamic factor model in time series analysis: the multivariate local level model with common trends; see Harvey and Koopman (1997). A Bayesian analysis of three monthly US short-term interest rates is presented.