  This paper proposes a new Bayesian approach for analysing moment condition
models in the situation where the data may be contaminated by outliers. The
approach builds upon the foundations developed by Schennach (2005) who proposed
the Bayesian exponentially tilted empirical likelihood (BETEL) method,
justified by the fact that an empirical likelihood (EL) can be interpreted as
the nonparametric limit of a Bayesian procedure when the implied probabilities
are obtained from maximizing entropy subject to some given moment constraints.
Considering the impact that outliers are thought to have on the estimation of
population moments, we develop a new robust BETEL (RBETEL) inferential
methodology to deal with this potential problem. We show how the BETEL methods
are linked to the recent work of Bissiri, Holmes and Walker (2016) who propose
a general framework to update prior belief via a loss function. A controlled
simulation experiment is conducted to investigate the performance of the RBETEL
method. We find that the proposed methodology produces reliable posterior
inference for the fundamental relationships that are embedded in the majority
of the data, even when outliers are present. The method is also illustrated in
an empirical study relating brain weight to body weight using a dataset
containing sixty-five different land animal species.
