Design of vector autoregressive processes for invariant statistics

This paper discusses the Monte Carlo (MC) design of Gaussian Vector Au- toregressive processes (VAR) for the evalutation of invariant statistics. We focus on the case of cointegrated (CI) I(1) processes, linear and invertible trans- formations and CI rank likelihood ratio (LR) tests. It is found that all VAR of order 1 can be reduced to a system of independent or recursive subsystems, of computational dimension at most equal to 2. The results are applied to the indexing of the distribution of LR test statistics for CI rank under local alternatives. They are also extended to the case of VAR processes of higher order.