  Cointegration is an important concept in the analysis of non-stationary
time-series, giving conditions under which a collection of non-stationary
processes has an underlying stationary (cointegration) relationship. In this
paper we present the first fully Bayesian residual-based test for
cointegration, where we consider the whole space of possible cointegration
relationships when testing for the presence of cointegration. We first
demonstrate that such a test can be performed exactly in the case where the
residual process follows a first-order autoregressive process. We then extend
this test to include more complex residual processes, where we first consider a
suitable cointegration test-statistic and then leverage Bayesian sampling
techniques to perform the necessary inference. We empirically demonstrate that
our Bayesian approach attains a superior classification accuracy than existing
approaches, all of which use a point estimate of the cointegration relationship
in their test. Finally, we demonstrate our approach on some real world
financial time-series data.
