  In a linear regression model with random design, we consider a family of
candidate models from which we want to select a `good' model for prediction
out-of-sample. We fit the models using block shrinkage estimators, and we focus
on the challenging situation where the number of explanatory variables can be
of the same order as sample size and where the number of candidate models can
be much larger than sample size. We develop an estimator for the out-of-sample
predictive performance, and we show that the empirically best model is
asymptotically as good as the truly best model. Using the estimator
corresponding to the empirically best model, we construct a prediction interval
that is approximately valid and short with high probability, i.e., we show that
the actual coverage probability is close to the nominal one and that the length
of this prediction interval is close to the length of the shortest but
infeasible prediction interval. All results hold uniformly over a large class
of data-generating processes. These findings extend results of Leeb (2009),
where the models are fit using least-squares estimators, and of Huber (2013),
where the models are fit using shrinkage estimators without block structure.
