Ridge regression under dense factor augmented models

35 Pages Posted: 13 Nov 2020 Last revised: 5 Aug 2022

See all articles by Yi He

Yi He

University of Amsterdam - Amsterdam School of Economics (ASE); Tinbergen Institute

Date Written: July 20, 2022


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

Keywords: High-dimensional linear model; spectral analysis; Tikhonov regularization; mixed-effects model; cross-validation

JEL Classification: C51, C55, C22

Suggested Citation

He, Yi, Ridge regression under dense factor augmented models (July 20, 2022). Available at SSRN: https://ssrn.com/abstract=3699669 or http://dx.doi.org/10.2139/ssrn.3699669

Yi He (Contact Author)

University of Amsterdam - Amsterdam School of Economics (ASE) ( email )

Roetersstraat 11
Amsterdam, North Holland 1018 WB

HOME PAGE: http://yihe.nl

Tinbergen Institute ( email )

Burg. Oudlaan 50
Rotterdam, 3062 PA

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