Bayesian Estimation of a Covariance Matrix: Application for Asset and Liabiliy Management

Many authors have considered the problem of estimating a covariance matrix in small samples. In this framework the sample covariance matrix is not robust, the solution is to impose some ad hoc structure on the covariance matrix to force it to be well-conditioned. This method is known as shrinkage. Here we approach the problem with an hierarchical bayesian perspective : we propose hierarchical priors for the covariance matrix to shrink toward diagonality. This approach draws on the works of M.J. Daniels and R.E. Kass (Nonconjugate bayesian estimation of covariance matrices and its use in hierarchical models, 1999) but we have searched to avoid any influence of the hyperparameters on the inference. We use an Inverse Wishart prior and we place flat priors on the logarithm of its hyperparameters. The problem of estimating is under the Stein’s loss function and a Markov Chain Monte Carlo sampling scheme is used to implement posterior inference in the proposed model. A simulation study allows us to assess the performance of the estimator in terms of small-sample risk. Our bayesian estimator is then applied to a real longitudinal example from portfolio selection, in which the dimension of the covariance matrix is large relative to the sample size.