We present several Markov chain Monte Carlo simulation methods that have been widely used in recent years in econometrics and statistics. Among these is the Gibbs sampler, which has been of particular interest to econometricians. Although the paper summarizes some of the relevant theoretical literature, its emphasis is on the presentation and explanation of applications to important models that are studied in econometrics. We include a discussion of some implementation issues, the use of the methods in connection with the EM algorithm, and how the methods can be helpful in model specification questions. Many of the applications of these methods are of particular interest to Bayesians, but we also point out ways in which frequentist statisticians may find the techniques useful.

Likelihood inference with hierarchical models is often complicated by the fact that the likelihood function involves intractable integrals. Numerical integration (e.g. quadrature) is an option if the dimension of the integral is low but quickly becomes unreliable as the dimension grows. An alternative approach is to approximate the intractable integrals using Monte Carlo averages. Several dierent algorithms based on this idea have been proposed. In this paper we discuss the relative merits of simulated maximum likelihood, Monte Carlo EM, Monte Carlo Newton-Raphson and stochastic approximation. Key words and phrases : Eciency, Monte Carlo EM, Monte Carlo Newton-Raphson, Rate of convergence, Simulated maximum likelihood, Stochastic approximation All three authors partially supported by NSF Grant DMS-00-72827. 1 1

Hierarchical models (HMs) provide a flexible framework for modeling data. The ongoing development of techniques like the EM algorithm and Markov chain Monte Carlo has enabled statisticians to make use of increasingly more complicated HMs over the last few decades. In this article, we consider Bayesian and frequentist versions of a general, two-stage HM, and describe several examples from the literature that illustrate its versatility. Some key aspects of the computational techniques that are currently used in conjunction with this HM are then examined in the context of McCullagh and Nelder's (1989) salamander data. Several areas that are ripe for new research are identified.