To estimate causal effects from observational data, an applied researcher must impose beliefs. The instrumental variables exclusion restriction, for example, represents the belief that the instrument has no direct effect on the outcome of interest. Yet beliefs about instrument validity do not exist in isolation. Applied researchers often discuss the likely direction of selection and the potential for measurement error in their articles but lack formal tools for incorporating this information into their analyses. Failing to use all relevant information not only leaves money on the table
it runs the risk of leading to a contradiction in which one holds mutually incompatible beliefs about the problem at hand. To address these issues, we first characterize the joint restrictions relating instrument invalidity, treatment endogeneity, and non-differential measurement error in a workhorse linear model, showing how beliefs over these three dimensions are mutually constrained by each other and the data. Using this information, we propose a Bayesian framework to help researchers elicit their beliefs, incorporate them into estimation, and ensure their mutual coherence. We conclude by illustrating our framework in a number of examples drawn from the empirical microeconomics literature.