# Alternative normalisation to triangular restriction on cointegrating vectors

Johansen recommends that cointegrating vectors must be normalised for inference making purposes. All software packages use triangular normalisation of the cointegrating vectors i.e. the top $r$ by $r$ block of the estimated $\beta$ matrix is an identity matrix. Usually, this is achieved by deriving the row echelon form of the estimated long-run equations (cointegrating equations matrix).

Does any one know of any other normalisation forms? or methods to formulate different normalisation? Thanks

## 1 Answer

The issue is not so much normalization as it is identification of the parameters of the co-integration vectors. The estimated co-integration vectors define the co-integration space, but many other sets of vectors span this same space.

In the case where you have a single co-integration vector, normalization is sufficient for identification. This is not the case when you have more than one vector in which case you need to impose restrictions, the most common set of identifying restrictions being the triangular normalization you mention.

Johansen (1995, section 2 theorem 3) provides a set of algebraic conditions which are fairly easy to compute (but tedious to type) to verify that a set of restrictions imposed on the co-integration vectors is identifying.

It is also possible to identify the co-integration vectors by imposing restrictions on the adjustment matrix ($\alpha$). This fairly recent overview by Johansen might be useful.

Reference: Johansen, Søren. "Identifying restrictions of linear equations with applications to simultaneous equations and cointegration." Journal of econometrics 69.1 (1995): 111-132.