Tractable Bayesian Inference for Unidentified Linear Regression Models

In this paper, I propose a tractable approach to Bayesian inference in linear regression models for which the standard exogeneity assumption does not hold. By specifying a beta prior for the squared correlation between an error term and regressor, I demonstrate that the implied prior for a bias parameter is $t$-distributed. If the posterior distribution for the identified regression coefficient is normal, this implies that the posterior distribution for the unidentified treatment effect is the convolution of a normal distribution and a $t$-distribution. This result is closely related to the literatures on unidentified and partially identified regression models, imperfect instrumental variables, and sensitivity analysis