How I use cross validation to estimate parameter for cointegration?

Question

I am evaluating cointegration in pair of stocks. I make an Ordinary Least Square regression, and than I test if the residuals are stationary using the Dick Fuller Statistic on the residual. I will estimate a Vector Error Correction Model and I am also considering testing the Gaussian Assumptions as recomended by IrishStat in my other question.( Is is correct to compare t-statistics of different pairs of cointegrated timeseries?) I would like to use cross validation but I don´t know how. If I make k-fold cross validation, I will have k different sets of parameter. How may I evaluate the best model? I would like to have a mean model, without overfitting, that explain the long run relationship of the cointegrated pair. How can I build this mean model? I think that it is arbitrary to use the best result of the k-fold cross validation. What is your opinion? Am I overfiting if I choose from the k-folds results the model that is more stationary? May I consider testing the performance of the models in the role dataset? Thank you!