Bayesian Limited Information Analysis Revisited

It is shown that uniform priors give sharp, explosive posteriors in an underidentified simultaneous equations model (SEM ). It is also shown that, for a standard class of noninformative prior densities, the posterior density of the parameters of a single structural equation derived in a limited information framework, is a so-called ratio-form poly — t density if, and only if, the prior degrees of freedom parameter has the value suggested by Dreze ( 1976 ). Three representations of the incomplete simultaneous equations model are investigated and compared. Conditions are given under which the prior specification is invariant under the different formulations of the model. Posterior densities of the parameters of interest are derived, in particular, the class of poly —matrix — t densities. The use of the distribution theoretic results for the analysis of overidentification and exogeneity in a Bayesian framework is, briefly, discussed.