The Asymptotic Distribution of Likelihood Ratio Test Statistics for Cointegration in Unstable Vector Autoregressive Processes

When analysing cointegration in vector autoregressive models it is usually assumed that (i) the number of cointegrating relations is not smaller than what is tested for, (ii) the number of unit roots equals the number of common stochastic trends, and (iii) the remaining characteristic roots of the time series are stationary roots. Condition (iii) could be violated in data from hyper-inflationary economies and in connection with seasonally integrated data while condition (ii) is not satisfied by processes which are integrated or order two. Here it is proved that condition (ii) and (iii) are redundant when testing linear restrictions on the cointegrating vector.