I am using elastic net regression to build explanatory models of clinical outcome measures using mutually correlated predictors. In elastic net regression, does choosing the regularisation parameter (Lambda) by cross-validation, in order to minimise MSE, prevent overfitting? Do the degrees of freedom provide an indication of overfitting when N is small?
More often it reduces over-fitting by shrinking the coefficient estimates. But the lowest cross-validated MSE estimate over the regularization parameter is optimistic when taken as an estimate of the out-of-sample performance of the model using that value of the regularization parameter; though not usually by much, as the optimization is very constrained. Use cross-validation or bootstrap validation of the whole procedure to estimate how much.