Posterior Simulators in Econometrics

John Geweke
University of Minnesota and Federal Reserve Bank in Minneapolis
Geweke@bayes.econ.umn.edu
http://www.econ.umn.edu/~jfgftp

Abstract

Economics is the discipline of using data to revise beliefs about economic issues. In Bayesian econometrics, the revision is conducted in accordance with the laws of probability, conditional on what has been observed. The normative appeal of Bayesian econometrics is the same as that of expected utility maximization and Bayesian learning, the dominant paradigms in economic theory. The questions that econometrics ultimately addresses are similar to those faced by economic agents in models, as well. Given the observed data, what decisions should be made? After bringing data to bear on two alternative models, how is their relative plausibility changed? Any survey of the introductory and concluding sections of papers in the academic literature should provide more examples and illustrate the process of formally or informally updating beliefs.

Until quite recently, applied Bayesian econometrics was undertaken largely by those primarily concerned with contributing to the theory, and the proportion of applied work that was formally Bayesian was rather small. There are several reasons for this. First, Bayesian econometrics demands both a likelihood function and a prior distribution, whereas non-Bayesian methods do not. Second, the subjective prior distribution has to be defended, and if the reader (or worse, the editor) does not agree, then the work may be ignored. Third, most posterior moments can't be obtained anyway because the requisite integrals can't be evaluated.

The development of posterior simulators in the last decade has revised beliefs about the foregoing three propositions held by many econometricians who have followed these developments closely. The purpose of this paper to to convey these innovations and their significance for applied econometrics, to econometricians who have not followed the relevant mathematical and applied literature. There are four substantive sections. One section reviews aspects of Bayesian inferences essential to understanding the implications of posterior simulators for Bayesian econometrics. Another section describes these simulators and provides the essential convergence results. Implications of these procedures for some selected econometric models are drawn in a third section. This is done to indicate the range of tasks to which posterior simulators are well suited, rather than to provide a representative survey of the recent Bayesian econometric literature. Finally, the paper turns to some implications for model comparison, and for communication between those who do applied work and their audiences, that are beginning to emerge from the use of posterior simulators in Bayesian econometrics.

The paper concludes that, with breakthroughs in importance and Markov chain sampling, the statement that most posterior moments are unobtainable because of technical difficulties with integration is now false. Econometrics models in which any posterior moment of interest that exists cannot be obtained using a posterior simulator are now the exception, not the rule. In the past two years there have emerged important cases in which the posterior moments are more easily and reliably obtained than are non-Bayesian estimates. This is especially the case for models with latent variables.

The implications of posterior simulators for model comparison and the communication of results bear on the subjectivity of the prior distribution. It is now the case that the readers and clients need not be passive and can conveniently take a role in the specification of econometrics models and their application. The reader is free to explore posterior moments of his choice and examine the implications of revisions of the investigator's prior distribution for those moments. Indeed, the investigator can choose her prior to facilitate this process.

If exploration of priors by clients becomes commonplace, then questions about the impact of subjective choices made by the econometrician shifts from the prior distribution to the functional form of the data distribution. This choice is made subjectively in Bayesian and many non-Bayesian procedures alike, and when it is not made explicitly in non-Bayesian procedures, then implicit restrictions on functional form exist in the assumed applicability of a central limit theorem. Alternative functional forms for data distributions can be compared using Bayes factors; no such general comparison is possible using non-Bayesian methods. Within the past two years, reliable methods of approximation for the marginalized likelihoods that constitute Bayes factors have become available, and rapid further progress is currently being made.