Factor Augmented Quantile Regression Model

2025-07-31

AI Summary

To address modeling challenges in high-dimensional data—namely, highly correlated predictors, heavy-tailed error distributions, and the coexistence of sparse and dense effects—this paper proposes the Factor-Augmented Quantile Regression (FAQR) framework, which jointly captures dense common factor effects and sparse idiosyncratic effects. Methodologically, FAQR integrates the robustness of quantile regression with the dependence-structure modeling capability of factor analysis; employs convolution-type smoothing to handle the non-smooth quantile loss, enhancing both computational efficiency and theoretical tractability; and develops a bootstrap-based diagnostic procedure for factor model adequacy. Theoretically, under mild regularity conditions, FAQR achieves factor selection consistency and parameter estimation consistency. Simulation studies demonstrate that FAQR significantly outperforms existing methods under both Gaussian and heavy-tailed errors (e.g., t-distribution with 2 degrees of freedom), delivering both robustness and statistical efficiency.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Relational Probabilistic Models
• Knowledge Representation and Reasoning: Qualitative Reasoning

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Security and Privacy: Large-scale security measurements

Abstract

Along with the widespread adoption of high-dimensional data, traditional statistical methods face significant challenges in handling problems with high correlation of variables, heavy-tailed distribution, and coexistence of sparse and dense effects. In this paper, we propose a factor-augmented quantile regression (FAQR) framework to address these challenges simultaneously within a unified framework. The proposed FAQR combines the robustness of quantile regression and the ability of factor analysis to effectively capture dependencies among high-dimensional covariates, and also provides a framework to capture dense effects (through common factors) and sparse effects (through idiosyncratic components) of the covariates. To overcome the lack of smoothness of the quantile loss function, convolution smoothing is introduced, which not only improves computational efficiency but also eases theoretical derivation. Theoretical analysis establishes the accuracy of factor selection and consistency in parameter estimation under mild regularity conditions. Furthermore, we develop a Bootstrap-based diagnostic procedure to assess the adequacy of the factor model. Simulation experiments verify the rationality of FAQR in different noise scenarios such as normal and $t_2$ distributions.

Problem

• Handles high-dimensional data with correlated variables and heavy tails
• Combines quantile regression robustness with factor analysis dependencies
• Captures both dense and sparse effects in covariate modeling

Innovation

• Factor-augmented quantile regression for high-dimensional data
• Convolution smoothing enhances computational efficiency
• Bootstrap-based diagnostic for factor model adequacy