When estimating a Vector Error Correction (VEC) model in EViews, the resulting output always shows the cointegrating relations together with standard errors and t-values for the included variables, and I was wondering if and how these numbers matter for the results?

In my understanding the presence of cointegration among two or more variables is determined based on the Johansen Cointegration Trace/Eigenvalue test, and the actual cointegrating relation(s) are then retrieved by estimating a VEC model of the variables. Following this procedure, however, all variables included in the VEC estimation enter the cointegrating relation, even though some of them may have no long-term relation with the other variables whatsoever, and the SE/t-values of the variables in the cointegrating relation change depending on the ordering specified. Do SE/t-values actually play a role in determining the correct cointegrating relation(s)?