Abstract
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.
Highlights
Research on Bayesian analysis of the simultaneous equations models addresses a problem, raised initially by Maddala (1976), and recognized as related to the problem of local nonidentification when diffuse/flat priors are used in traditional Bayesian analysis, e.g., Drèze (1976); Drèze and Morales (1976), and Drèze and Richard (1983).1 In this paper, we examine the approaches developed byChao and Phillips (1998, hereafter CP), Geweke (1996), Kleibergen and van Dijk (1998, hereafterKVD), and Zellner (1998)
The advantage of this algorithm is that it 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 on convergence are weighted to generate drawings from the target density in a Metropolis–Hastings (M–H) algorithm
We note that minimum expected loss (MELO), LIML-Gibbs sampling (GS)-Mode, and KVD-Mode or KVD-Median are median-biased in the direction of ρ
Summary
Research on Bayesian analysis of the simultaneous equations models addresses a problem, raised initially by Maddala (1976), and recognized as related to the problem of local nonidentification when diffuse/flat priors are used in traditional Bayesian analysis, e.g., Drèze (1976); Drèze and Morales (1976), and Drèze and Richard (1983). In this paper, we examine the approaches developed byChao and Phillips (1998, hereafter CP), Geweke (1996), Kleibergen and van Dijk (1998, hereafterKVD), and Zellner (1998). While KVD focused mainly on resolving the problem of local nonidentification, CP explored further the consequences of using a Jeffreys prior. Geweke (1996), being aware of the potential problem of local nonidentification, suggests a shrinkage prior such that the posterior density is properly defined for each parameter. In another approach, Zellner (1998) suggested a finite sample Bayesian method of moments (BMOM) procedure based on given data without specifying a likelihood function or introducing any sampling assumptions
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