Suppose that we have the regression model
$$Y(t)=\alpha +\beta_1X_1(t)+ \cdots +\beta_nX_n(t)+\epsilon(t)$$
One approach to fitting this model is to use OLS. If the predictor variables $X_1(t),\ldots,X_n(t)$ are correlated then we are subject to statistical colinearity distortion and OLS can give an overfit model. These problems can be avoided by using the LASSO or ridge regression.
My question is, what happens when the predictor variables $X_1(t),\ldots,X_n(t)$ are cointegrated? What kinds of problems are introduced by cointegrated predictors and what are the best methods to use to get a reliable fit?