Reflections on the Probability Space Induced by Moment Conditions with Implications for Bayesian Inference

Abstract

Often a structural model implies that certain moment functions expressed in terms of data and model parameters follow a distribution. An assertion that moment functions follow a distribution logically implies a distribution on the arguments of the moment functions. This fact would appear to permit Bayesian inference on model parameters. The classic example is an assertion that the sample mean centered at a parameter and scaled by its standard error has Student’s t-distribution followed by an assertion that the sample mean plus and minus a critical value times the standard error is a Bayesian credibility interval for the parameter. This article studies the logic of such assertions. The main finding is that if the moment functions have one of the properties of a pivotal, then the assertion of a distribution on moment functions coupled with a proper prior does permit Bayesian inference. Without the semi-pivotal condition, the assertion of a distribution for moment functions either partially or completely specifies the prior. In this case, Bayesian inference may or may not be practicable depending on how much of the distribution of the constituents remains indeterminate after imposition of a noncontradictory prior. An asset pricing example that uses data from the U.S. economy illustrates the ideas.