A Comparison of Some Recent Bayesian Procedures for Simultaneous Equation Models Using Markov Chain Monte Carlo *

Chuanming Gao and Kajal Lahiri

Recently, Chao and Phillips (1998, CP) and Kleibergen and van Dijk (1998, KVD) developed two approaches to estimate simultaneous equations models in a Bayesian framework. In this paper, we examine these apporaches as well as the one by Geweke (1996) which may mitigate or avaoid the consequences of local non-identification. Since the posterior densities and their conditionals resulting from CP and KVD approaches are non-standard, we propose a ‘Gibbs within M-H’ algorithm, which only requires the availability of the conditional densities from the candidate generating density. These conditional densities are used in a Gibbs sampler to simulate the candidate generating density, whose drawings, after convergence, are then weighted to generate drawings from the target density in a Metopolis-Hastings algorithm. Comparing the performance of these Bayesian approaches using MCMC, together with various convergence diagnostic analysis [e.g. Cowles and Carlin (1996), and Brooks and Roberts (1999)] on MCMC output, adds further to the understanding of many issues involved in the estimation of SEMs. Our simulationstudy focuses on situations like the presence of weak instruments or weak simultaneity, where classical approach is particularly bad. For the purpose of comparison, OLS, 2SLS, classical LIML, and Zellner’s Extended MELO and BMOM estimators (1998) are also compared. Overall, we find the KVD approach shows little merit, whereas CP and Geweke perform well under different situations.