Priors, Posterior Odds and Lagrange Multiplier Statistics in Bayesian Analyses of Cointegration

Using the standard linear model as a base, a unified theory of Bayesian Analyses of Cointegration Models is constructed. This is achieved by defining (natural conjugate) priors in the linear model and using the implied priors for the cointegration model. Using these priors, posterior results for the cointegration model are obtained using a Metropolis-Hasting sampler. To compare the cointegration models mutually and with the vector autoregressive model under stationarity, we use two strategies. The first strategy uses the Bayesian interpretation of a Lagrange Multiplier statistic. The second strategy compares the models using prior and posterior odds ratios. The latter enables us to compute prior and posterior distributions over the cointegration rank and shows close resemblance with the posterior information criterium from Phillips and Ploberger (1996). To show the applicability of the derived theory, the constructed procedures are applied to data from Johansen and Juselius (1990) and a few simulated data sets.