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A robust and scalable framework for high-dimensional volatility estimation

Kejun Chen, Yuchang Lin, Qianqian Zhu

TL;DR

The proposed approach employs data truncation to ensure robustness against heavy-tailed distributions and utilizes a regularized least squares method for efficient optimization in high-dimensional settings by leveraging an equivalent VAR representation of the BEKK-ARCH model.

Abstract

This paper introduces a robust and computationally efficient estimation framework for high-dimensional volatility models in the BEKK-ARCH class. The proposed approach employs data truncation to ensure robustness against heavy-tailed distributions and utilizes a regularized least squares method for efficient optimization in high-dimensional settings. This is achieved by leveraging an equivalent VAR representation of the BEKK-ARCH model. Non-asymptotic error bounds are established for the resulting estimators under heavy-tailed regime, and the minimax optimal convergence rate is derived. Moreover, a robust BIC and a Ridge-type estimator are introduced for selecting the model order and the number of BEKK components, respectively, with their selection consistency established under heavy-tailed settings. Simulation studies demonstrate the finite-sample performance of the proposed method, and two empirical applications illustrate its practical utility. The results show that the new framework outperforms existing alternatives in both computational speed and forecasting accuracy.

A robust and scalable framework for high-dimensional volatility estimation

TL;DR

The proposed approach employs data truncation to ensure robustness against heavy-tailed distributions and utilizes a regularized least squares method for efficient optimization in high-dimensional settings by leveraging an equivalent VAR representation of the BEKK-ARCH model.

Abstract

This paper introduces a robust and computationally efficient estimation framework for high-dimensional volatility models in the BEKK-ARCH class. The proposed approach employs data truncation to ensure robustness against heavy-tailed distributions and utilizes a regularized least squares method for efficient optimization in high-dimensional settings. This is achieved by leveraging an equivalent VAR representation of the BEKK-ARCH model. Non-asymptotic error bounds are established for the resulting estimators under heavy-tailed regime, and the minimax optimal convergence rate is derived. Moreover, a robust BIC and a Ridge-type estimator are introduced for selecting the model order and the number of BEKK components, respectively, with their selection consistency established under heavy-tailed settings. Simulation studies demonstrate the finite-sample performance of the proposed method, and two empirical applications illustrate its practical utility. The results show that the new framework outperforms existing alternatives in both computational speed and forecasting accuracy.
Paper Structure (32 sections, 17 theorems, 170 equations, 6 figures, 5 tables)

This paper contains 32 sections, 17 theorems, 170 equations, 6 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Methodology
  3. BEKK-ARCH model and its equivalent form
  4. Estimation
  5. Recovering BEKK-ARCH model from its vech form
  6. Implementation Issues
  7. Algorithm
  8. Model selection
  9. Tuning parameter selection
  10. Theoretical Analysis
  11. Error bounds for the regularized LSE
  12. Error bounds for estimating BEKK coefficient matrices
  13. Model selection consistency
  14. Simulation
  15. Vech BEKK model estimation
  16. ...and 17 more sections

Key Result

Proposition 1

Let $\{\mathbf{B}_{ik}\}_{k=1}^{K_i^B} \subset \mathbb{R}^{N\times N}$ be the collection of coefficient matrices at lag $i$ in the BEKK model eq:BEKK-ARCH. Then there exists an integer $K_i^A \leq K_i^B$ and a set of matrices $\{\mathbf{A}_{ik}\}_{k=1}^{K_i^A} \subset \mathbb{R}^{N\times N}$ that ar

Figures (6)

  • Figure 1: Estimation errors in the $\ell_{2,\infty}$-norm and Frobenius norm for the regularized LSE, with and without truncation, versus sample size $T$ for $(N,s)=(20,3)$.
  • Figure 2: Estimation errors in the $\ell_{2,\infty}$-norm and Frobenius norm for the regularized LSE, with and without truncation, versus $T$ for $(N,s)=(100,10)$.
  • Figure 3: Estimation errors in the Frobenius norm for three estimators of conditional covariance matrix $\bm{\Sigma}_t$, versus sample size $T$ for $(N,s)=(20,3)$.
  • Figure 4: Estimation errors in the Frobenius norm for three estimators of conditional covariance matrix $\bm{\Sigma}_t$, versus $T$ for $(N,s)=(100,10)$.
  • Figure 5: Estimation errors of $\widehat{\bm{\Omega}}$ and $\widehat{\mathbf{A}}_{ik}$ in the Frobenius norm, versus sample size $T$ for $(N,s)=(20,3)$.
  • ...and 1 more figures

Theorems & Definitions (32)

  • Proposition 1: Orthogonalization invariance of BEKK coefficient matrices
  • Theorem 1: Upper bounds for regularized LSE
  • Theorem 2: Minimax lower bound for regularized LSE
  • Corollary 1: BEKK-ARCH Upper Bound
  • Theorem 3: Selection consistency of $\widehat{p}$
  • Theorem 4: Selection consistency of $\widehat{\mathcal{K}}_p$
  • Definition A.1: Localized Restricted Eigenvalue
  • Lemma B.1
  • Lemma B.2
  • Lemma B.3: $\ell_1$-Cone
  • ...and 22 more