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High-Dimensional Dynamic Covariance Models with Random Forests

Shuguang Yu, Fan Zhou, Yingjie Zhang, Ziqi Chen, Hongtu Zhu

TL;DR

This work introduces Forest-based Dynamic Covariance Models (FDCMs), a fully nonparametric framework for estimating high-dimensional conditional covariance matrices $\boldsymbol{\Sigma}(\boldsymbol{U})$ as functions of multiple conditioning covariates $\boldsymbol{U}$. By leveraging honest random forests to derive data-driven weights for estimating $\mathbb{E}(\boldsymbol{Y}\boldsymbol{Y}^\top|\boldsymbol{U})$ and $\mathbb{E}(\boldsymbol{Y}|\boldsymbol{U})$, and applying a sparsity-promoting thresholding operator, the method yields a positive-definite, sparse estimator that remains uniformly consistent as both sample size and dimension grow. The authors provide rigorous uniform convergence results for the estimator and its inverse, along with sparsity-pattern recovery guarantees, under mild regularity conditions. Simulations demonstrate substantial accuracy gains over static and kernel-based methods, especially when multiple covariates drive the dynamics, and a real-data study on NYSE stocks conditioned on Fama–French factors shows improved out-of-sample portfolio performance. Overall, the paper advances high-dimensional covariance estimation by integrating random forests with dynamic, multivariate conditioning and robust theoretical guarantees, offering a practical tool for finance and related fields.

Abstract

This paper introduces a novel nonparametric method for estimating high-dimensional dynamic covariance matrices with multiple conditioning covariates, leveraging random forests and supported by robust theoretical guarantees. Unlike traditional static methods, our dynamic nonparametric covariance models effectively capture distributional heterogeneity. Furthermore, unlike kernel-smoothing methods, which are restricted to a single conditioning covariate, our approach accommodates multiple covariates in a fully nonparametric framework. To the best of our knowledge, this is the first method to use random forests for estimating high-dimensional dynamic covariance matrices. In high-dimensional settings, we establish uniform consistency theory, providing nonasymptotic error rates and model selection properties, even when the response dimension grows sub-exponentially with the sample size. These results hold uniformly across a range of conditioning variables. The method's effectiveness is demonstrated through simulations and a stock dataset analysis, highlighting its ability to model complex dynamics in high-dimensional scenarios.

High-Dimensional Dynamic Covariance Models with Random Forests

TL;DR

This work introduces Forest-based Dynamic Covariance Models (FDCMs), a fully nonparametric framework for estimating high-dimensional conditional covariance matrices as functions of multiple conditioning covariates . By leveraging honest random forests to derive data-driven weights for estimating and , and applying a sparsity-promoting thresholding operator, the method yields a positive-definite, sparse estimator that remains uniformly consistent as both sample size and dimension grow. The authors provide rigorous uniform convergence results for the estimator and its inverse, along with sparsity-pattern recovery guarantees, under mild regularity conditions. Simulations demonstrate substantial accuracy gains over static and kernel-based methods, especially when multiple covariates drive the dynamics, and a real-data study on NYSE stocks conditioned on Fama–French factors shows improved out-of-sample portfolio performance. Overall, the paper advances high-dimensional covariance estimation by integrating random forests with dynamic, multivariate conditioning and robust theoretical guarantees, offering a practical tool for finance and related fields.

Abstract

This paper introduces a novel nonparametric method for estimating high-dimensional dynamic covariance matrices with multiple conditioning covariates, leveraging random forests and supported by robust theoretical guarantees. Unlike traditional static methods, our dynamic nonparametric covariance models effectively capture distributional heterogeneity. Furthermore, unlike kernel-smoothing methods, which are restricted to a single conditioning covariate, our approach accommodates multiple covariates in a fully nonparametric framework. To the best of our knowledge, this is the first method to use random forests for estimating high-dimensional dynamic covariance matrices. In high-dimensional settings, we establish uniform consistency theory, providing nonasymptotic error rates and model selection properties, even when the response dimension grows sub-exponentially with the sample size. These results hold uniformly across a range of conditioning variables. The method's effectiveness is demonstrated through simulations and a stock dataset analysis, highlighting its ability to model complex dynamics in high-dimensional scenarios.
Paper Structure (10 sections, 3 theorems, 25 equations, 2 figures, 4 tables, 1 algorithm)

This paper contains 10 sections, 3 theorems, 25 equations, 2 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related work
  3. Notation
  4. Methodology
  5. Theory
  6. Simulation Studies
  7. Real Data Analysis
  8. The dynamic covariance models
  9. The comparisons of minimum-variance portfolios
  10. Conclusion

Key Result

Theorem 1

Under Conditions (a)--(e), suppose that $s_{\lambda}$ is a shrinkage operator. Uniformly on $\mathcal{U}(q,c_0(p),M_1;[0,1]^d)$, if $\lambda_n(\boldsymbol u)=M(\boldsymbol u)\left(\sqrt{s\left\{\log(n)+\log(p)\right\}/n}+s^{-\frac{1}{2}\frac{\log(1-\omega)}{\log(\omega)}\frac{\pi}{d}}\right)$ and $s where $M(\boldsymbol u)$ depending on $\boldsymbol u\in[0,1]^d$ is large enough and $\sup_{\boldsym

Figures (2)

  • Figure 1: Returns on the Fama-French five factors as functions of time. The horizontal axis from 1 to 100 corresponds to the first 100 trading days from January 2, 2014, to December 29, 2023, arranged in chronological order.
  • Figure 2: Selected entries of the correlation matrix as functions of Fama-French five-factor return vector using (\ref{['covariancehat']}). The horizontal axis from $\boldsymbol U_1$ to $\boldsymbol U_{100}$ corresponds to returns on the Fama-French five-factor return vector at the first 100 trading days from January 2, 2014, to December 29, 2023 and $\text{corr}(j,r)$ denotes the $(j,r)$-th element of the sample correlation matrix.

Theorems & Definitions (3)

  • Theorem 1: Uniform consistency of the estimated dynamic matrix
  • Theorem 2: Uniform consistency in estimating the sparsity pattern
  • Theorem 3