We explore Bayesian inference of a multivariate linear regression model with use of a flexible prior for the covariance structure.\nThe commonly adopted Bayesian setup involves the conjugate prior,multivariate normal distribution for the regression coefficients\nand inverse Wishart specification for the covariance matrix. Here we depart from this approach and propose a novel Bayesian\nestimator for the covariance. A multivariate normal prior for the unique elements of the matrix logarithm of the covariance matrix\nis considered. Such structure allows for a richer class of prior distributions for the covariance, with respect to strength of beliefs\nin prior location hyperparameters, as well as the added ability, to model potential correlation amongst the covariance structure.\nThe posterior moments of all relevant parameters of interest are calculated based upon numerical results via a Markov chain Monte\nCarlo procedure. The Metropolis-Hastings-within-Gibbs algorithm is invoked to account for the construction of a proposal density\nthat closely matches the shape of the target posterior distribution. As an application of the proposed technique, we investigate a\nmultiple regression based upon the 1980 High School and Beyond Survey.
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