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We propose a robust Bayesian method for economic models that can be rejected by some data distributions. The econometrician starts with a refutable structural assumption which can be written as the intersection of several assumptions. To avoid the assumption refutable, the econometrician first takes a stance on which assumption $j$ will be relaxed and considers a function $m_j$ that measures the deviation from the assumption $j$. She then specifies a set of prior beliefs $\Pi_s$ whose elements share the same marginal distribution $\pi_{m_j}$ which measures the likelihood of deviations from assumption $j$. Compared to the standard Bayesian method that specifies a single prior, the robust Bayesian method allows the econometrician to take a stance only on the likeliness of violation of assumption $j$ while leaving other features of the model unspecified. We show that many frequentist approaches to relax refutable assumptions are equivalent to particular choices of robust Bayesian prior sets, and thus we give a Bayesian interpretation to the frequentist methods. We use the local average treatment effect ($LATE$) in the potential outcome framework as the leading illustrating example.

This paper studies the semi-parametric identification and estimation of a rational inattention model with Bayesian persuasion. The identification requires the observation of a cross-section of market-level outcomes. The empirical content of the model can be characterized by three moment conditions. A two-step estimation procedure is proposed to avoid computation complexity in the structural model. In the empirical application, I study the persuasion effect of Fox News in the 2000 presidential election. Welfare analysis shows that persuasion will not influence voters with high school education but will generate higher dispersion in the welfare of voters with a partial college education and decrease the dispersion in the welfare of voters with a bachelors degree.

This paper studies the identification, estimation, and hypothesis testing problem in complete and incomplete economic models with testable assumptions. Testable assumptions ($A$) give strong and interpretable empirical content to the models but they also carry the possibility that some distribution of observed outcomes may reject these assumptions. A natural way to avoid this is to find a set of relaxed assumptions ($\tilde{A}$) that cannot be rejected by any distribution of observed outcome and the identified set of the parameter of interest is not changed when the original assumption is not rejected. The main contribution of this paper is to characterize the properties of such a relaxed assumption $\tilde{A}$ using a generalized definition of refutability and confirmability. I also propose a general method to construct such $\tilde{A}$. A general estimation and inference procedure is proposed and can be applied to most incomplete economic models. I apply my methodology to the instrument monotonicity assumption in Local Average Treatment Effect (LATE) estimation and to the sector selection assumption in a binary outcome Roy model of employment sector choice. In the LATE application, I use my general method to construct a set of relaxed assumptions $\tilde{A}$ that can never be rejected, and the identified set of LATE is the same as imposing $A$ when $A$ is not rejected. LATE is point identified under my extension $\tilde{A}$ in the LATE application. In the binary outcome Roy model, I use my method of incomplete models to relax Roy's sector selection assumption and characterize the identified set of the binary potential outcome as a polyhedron.