  We study prediction intervals based on leave-one-out residuals in a linear
regression model where the number of explanatory variables can be large
compared to sample size. We establish uniform asymptotic validity (conditional
on the training sample) of the proposed interval under minimal assumptions on
the unknown error distribution and the high dimensional design. Our intervals
are generic in the sense that they are valid for a large class of linear
predictors used to obtain a point forecast, such as robust M-estimators,
James-Stein type estimators and penalized estimators like the LASSO. These
results show that despite the serious problems of resampling procedures for
inference on the unknown parameters, leave-one-out methods can be successfully
applied to obtain reliable predictive inference even in high dimensions.
