A distance measure between cointegration spaces

A distinguishing feature of cointegration models, and many other multivariate models, is that only spaces spanned by parameter vectors are identified. We point out that traditional distance measures, such as the Euclidean measure, are not reasonable to use when measuring distances between spaces. This point has been either missed or ignored in many simulation studies where inappropriate distance measures have been used. We propose a simple measure based on the idea that the space spanned by the orthogonal complement of a matrix lies as far away as possible from the space spanned by the matrix itself. Several properties of this measure are derived.