A Comparison of Some Bayesian and Classical Procedures for Simultaneous Equation Models with Weak Instruments

We compare the finite sample performance of a number of Bayesian and classical procedures for limited information simultaneous equations models with weak instruments by a Monte Carlo study. We consider Bayesian approaches developed by Chao and Phillips, Geweke, Kleibergen and van Dijk, and Zellner. Amongst the sampling theory methods, OLS, 2SLS, LIML, Fuller’s modified LIML, and the jackknife instrumental variable estimator (JIVE) due to Angrist et al. and Blomquist and Dahlberg are also considered. Since the posterior densities and their conditionals in Chao and Phillips and Kleibergen and van Dijk are nonstandard, we use a novel “Gibbs within Metropolis–Hastings” algorithm, which only requires the availability of the conditional densities from the candidate generating density. Our results show that with very weak instruments, there is no single estimator that is superior to others in all cases. When endogeneity is weak, Zellner’s MELO does the best. When the endogeneity is not weak and ρ ω 12 > 0 , where ρ is the correlation coefficient between the structural and reduced form errors, and ω 12 is the covariance between the unrestricted reduced form errors, the Bayesian method of moments (BMOM) outperforms all other estimators by a wide margin. When the endogeneity is not weak and β ρ < 0 ( β being the structural parameter), the Kleibergen and van Dijk approach seems to work very well. Surprisingly, the performance of JIVE was disappointing in all our experiments.