The authors examine the ability of the Johansen (1991) test to estimate the number of unit roots in high dimensional systems. They use data based Monte Carlo methods as a simple means of evaluating the validity of inference using asymptotic critical values. These simulations for a typical annual post-World War II dataset illustrate how the estimated number of unit roots change in a nonmonotone fashion with the dimension of the system, and with the number of lags in the VAR representation. The authors find that overparametrization in high dimensions is as bad as underparametrization. The Bayes information criteria outperforms the Akaike information criteria in their setup. Copyright 1996 by MIT Press.