Cointegration rank inference with stationary regressors in VAR models

The issue of including stationary explanatory variables is addressed in the vector autoregressive (VAR) model, when testing for cointegration rank. It is shown that simply in-cluding stationary explanatory variables as extra regressors will lead to nuisance parameters in the asymptotic distribution of the trace statistic for cointegration rank. The nuisance parame-ters are characterized as canonical correlations between the common trends (which in this case also involve the accumulated stationary explanatory process) and the accumulated innovations. Thus, in particular, the trace test is not similar, even asymptotically, and an alternative model is discussed, which will lead to nuisance-free rank determination. The alternative model is the extended VAR model where the cumulated explanatory variables enter the system in the error correction term. Possible loss in power due to overparametrization is briefly addressed and the proposed analysis is illustrated by an empirical example.