Computationally intensive methods for integration in econometrics

Until recently, inference in many interesting models was precluded by the requirement of high dimensional integration. But dramatic increases in computer speed, and the recent development of new algorithms that permit accurate Monte Carlo evaluation of high dimensional integrals, have greatly expanded the range of models that can be considered. This chapter presents the methodology for several of the most important Monte Carlo methods, supplemented by a set of concrete examples that show how the methods are used.Some of the examples are new to the econometrics literature. They include inference in multinomial discrete choice models and selection models in which the standard normality assumption is relaxed in favor of a multivariate mixture of normals assumption. Several Monte Carlo experiments indicate that these methods are successful at identifying departures from normality when they are present. Throughout the chapter the focus is on inference in parametric models that permit rich variation in the distribution of disturbances.The chapter first discusses Monte Carlo methods for the evaluation of high dimensional integrals, including integral simulators like the GHK method, and Markov Chain Monte Carlo methods like Gibbs sampling and the Metropolis-Hastings algorithm. It then turns to methods for approximating solutions to discrete choice dynamic optimization problems, including the methods developed by Keane and Wolpin, and Rust, as well as methods for circumventing the integration problem entirely, such as the approach of Geweke and Keane. The rest of the chapter deals with specific examples: classical simulation estimation for multinomial probit models, both in the cross sectional and panel data contexts; univariate and multivariate latent linear models; and Bayesian inference in dynamic discrete choice models in which the future component of the value function is replaced by a flexible polynomial.