Bayesian unit root tests and marginal likelihood

Unit root tests based on classical marginal likelihood are practically uniformly most powerful (Francke and de Vos, 2007). Bayesian unit root tests can be constructed that are very similar, however in the Bayesian analysis the classical size is determined by prior considerations. A fundamental difference remains the link between the implied size and the number of observations. To establish this correspondence, we get two intermediate results that may be important in a wider context. We prove that for inference on the covariance parameters in the general linear model classical and Bayesian versions of marginal likelihood are equivalent if Jeffreys’ independence priors are used. Further we show equivalence between classical and Bayesian tests under some monotonicity conditions.