This paper studies a generalized version of the relation between the sample correlation matrices of errors and their residuals. The results suggest confidence bands to select a proper multivariate time series model. Several simulations illustrate the theoretical findings. The literature of Environmental Kuznets Curve (EKC) received particular interest, however, the empirical research on the EKC has mainly used a reduced form approach. Therefore, the proposed techniques can be applied to identify the multivariate relation between economic growth, environmental degradation and population. A real data application is provided for the case of Sweden.