The paper presents a generalization of the Lindley and Smith (1972) linear hierarchical SUR model, concerning the prior probabilistic specification of the first level parameters of the model which are assumed to be partially exchangeable. More specifically, by extending the Diaconis and Ylvisaker (1985) approach to multivariate distributions, a finite mixture of natural conjugate probability densities is used as prior density for the parameters. This type of prior specification enables us to analytically perform the Bayesian updating of the regression coefficients and error variances relative to the first level of parameterization. We consider to different specifications for the prior on the parameters of the first level. The first one assumes a completely specified prior distribution, while a second level of parameterization is introduced in the last specification. In our view, the main features of our approach are greater flexibility in specifying the prior information and a notable
