Robust Bayesian Method for Refutable Models

We propose a robust Bayesian method for economic models that can be rejected under some data distributions. The econometrician starts with a structural assumption which can be written as the intersection of several assumptions, and the joint assumption is refutable. To avoid the model rejection, the econometrician first takes a stance on which assumption $j$ is likely to be violated and considers a measurement of the degree of violation of this assumption $j$. She then considers a (marginal) prior belief on the degree of violation $(\pi_{m_j})$: She considers a class of prior distributions $\pi_s$ on all economic structures such that all $\pi_s$ have the same marginal distribution $\pi_m$. Compared to the standard nonparametric Bayesian method that puts a single prior on all economic structures, the robust Bayesian method imposes a single marginal prior distribution on the degree of violation. As a result, the robust Bayesian method allows the econometrician to take a stance only on the likeliness of violation of assumption $j$. Compared to the frequentist approach to relax the refutable assumption, the robust Bayesian method is transparent on the econometrician's stance of choosing models. We also show that many frequentists' ways to relax the refutable assumption can be found equivalent to particular choices of robust Bayesian prior classes. We use the local average treatment effect (LATE) in the potential outcome framework as the leading illustrating example.