Jeffreys Prior Analysis of the Simultaneous Equations Model in the Case with n + 1 Endogenous Variables

This paper analyzes the behavior of posterior distributions under the Jeffreys prior in a simultaneous equations model. The case under study is that of a general limited information setup with n+1 endogenous variables. The Jeffreys prior is shown to give rise to a marginal posterior density which has Cauchy-like tails similar to that exhibited by the exact finite sample distribution of the corresponding LIML estimator. A stronger correspondence is established in the special case of a just-identified orthonormal canonical model, where the posterior density under the Jeffreys prior is shown to have the same functional form as the density of the finite sample distribution of the LIML estimator. The work here generalizes that of Chao and Phillips (1997), which gives analogous results for the special case of two endogenous variables.