The Kalman ﬁlter is sued to derive updating equations for the Bayesian data density in discrete time linear regression models with stochastic regressors. The implied “Bayes model” has time varying parameters and conditionally heterogeneous error variances. A sigma-ﬁnite “Bayes model” measure is given and used to produce a new model selection criterion (PIC) and objective posterior odds tests for sharp null hypotheses like the presence of a unit root. Simulation results and an empirical application are reported. The simulations show that the new model selection criterion “PIC” works very well and is generally superior to the Schwarz criterion BIC even in stationary systems.
Phillips, Peter C.B. and Ploberger, Werner, "Posterior Odds Testing for a Unit Root with Data-Based Model Selection" (1992). Cowles Foundation Discussion Papers. 1260.