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In a linear regression model with random design, we consider a family ofcandidate models from which we want to select a `good' model for predictionout-of-sample. We fit the models using block shrinkage estimators, and we focuson the challenging situation where the number of explanatory variables can beof the same order as sample size and where the number of candidate models canbe much larger than sample size. We develop an estimator for the out-of-samplepredictive performance, and we show that the empirically best model isasymptotically as good as the truly best model. Using the estimatorcorresponding to the empirically best model, we construct a prediction intervalthat is approximately valid and short with high probability, i.e., we show thatthe actual coverage probability is close to the nominal one and that the lengthof this prediction interval is close to the length of the shortest butinfeasible prediction interval. All results hold uniformly over a large classof 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.

Prediction out-of-sample using block shrinkage estimators: model selection and predictive inference