Tractable Bayesian inference for an unidentified simple linear regression model

In this paper, I propose a tractable approach to Bayesian inference in a simple linear regression model 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 regression models, imperfect instrumental variables, and sensitivity analysis.