Bayesian Posterior Distributions in Limited Information Analysis of the Simultaneous Equations Model Using the Jeffreys Prior

This paper studies the use of the Jeffreys' prior in Bayesian analysis of the simultaneous equations model (SEM). Exact representations are obtained for the posterior density of the structural coefficient beta in canonical SEM's with two endogenous variables. For the general case with m endogenous variables and an unknown covariance matrix, the Laplace approximation is used to derive an analytic formula for the same posterior density. Both the exact and the approximate formulas we derive are found to exhibit Cauchy-like tails analogous to comparable results in the classical literature on LIML estimation. Moreover, in the special case of a two-equation, just-identified SEM in canonical form, the posterior density of beta is shown to have the same infinite series representation as the density of the finite sample distribution of the corresponding LIML estimator. This paper also examines the occurrence of a nonintegrable asymptotic cusp in the posterior distribution of the reduced form parameter Phi, first documented in Kleibergen and van Dijk (1994). This phenomenon is explained in terms of the jacobian of the mapping from the structural model to the reduced form. This interpretation assists in understanding the success of the Jeffreys' prior in resolving this problem.