Limited information likelihood and Bayesian analysis

In this paper, we study how to embed the optimal generalized method of moments (GMM) estimate in a likelihood-based inference framework and the Bayesian framework. First, we derive a limited information likelihood (LIL) under some moment-based limited information available in GMM based on entropy theory of I-projection theory. Second, we study a limited information Bayesian framework in which the posterior is derived from the LIL and a prior. As the LIL enables us to incorporate GMM or related inference methods in the likelihood-based inference framework, it allows us a rich set of practical applications in the Bayesian framework in which the posterior is obtained from a likelihood and a prior. Our results are primarily large sample results as inference in the underlying GMM framework is usually justified in asymptotics. Investigation of large sample properties of the posterior derived from the LIL reveals an interesting relation between the Bayesian and the classical distribution theories.