Ridge Regression Under Dense Factor Augmented Models

This article establishes a comprehensive theory of the optimality, robustness, and cross-validation selection consistency for ridge regression under factor-augmented models with possibly dense idiosyncratic information. Using spectral analysis random matrices, we show that is asymptotically efficient in capturing both factor information by minimizing limiting predictive loss among entire class regularized estimators large-dimensional mixed-effects hypothesis. We derive an optimal penalty closed form prove bias-corrected k-fold procedure can adaptively select best large samples. extend to autoregressive many exogenous variables establish consistent using what-we-called double method. Our results allow nonparametric distributions for, heavy-tailed, martingale difference errors coefficients adapt cross-sectional temporal dependence structures predictors. demonstrate performance our simulated examples as well economic dataset. All proofs are available supplementary materials, which also includes more technical discussions remarks, extra simulation results, useful lemmas may be independent interest.