On Markov Chain Monte Carlo Methods

Abstract

This paper examines two Markov chain Monte Carlo methods that have been widely used in econometrics. An introductory exposition of the Metropolis algorithm and the Gibbs sampler is provided. These methods are used to simulate multivariate distributions. Many problems in Bayesian statistics can be solved by simulating the posterior distribution. Invariance condition is of importance, the proofs are given for both meth ods. We use finite Markov chains to explore and substantiate the methods. Several examples are provided to illustrate the applicability and efficiency of the Markov chain Monte Carlo methods. They include bivariate normal distribution with high correlation, bivaria­te exponential distribution, mixture of bivariate normals.