Testing for r versus r−1 cointegrating vectors

A parametric test for r versus r−1 cointegrating vectors is developed. The test exploits the fact that in a system of n I(1) variates the rth principal component is I(0) under the null but I(1) under the alternative. The statistic is parametric, is constructed using simple regression methods applied to principal components, follows a standard χ2 distribution and does not require normalisation restrictions on the cointegrating vectors. A Monte Carlo investigation indicates that providing the lag length in the pre-whitening procedure is chosen by means of nested significance tests, the test has good size and power properties in small samples.