  We propose a new prediction method for multivariate linear regression
problems where the number of features is less than the sample size but the
number of outcomes is extremely large. Many popular procedures, such as
penalized regression procedures, require parameter tuning that is
computationally untenable in such large-scale problems. We take a different
approach, motivated by ideas from simultaneous estimation problems, that
performs linear shrinkage on ordinary least squares parameter estimates. Our
approach is extremely computationally efficient and tuning-free. We show that
it can asymptotically outperform ordinary least squares without any structural
assumptions on the true regression coefficients and illustrate its good
performance in simulations and an analysis of single-cell RNA-seq data.
