Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

High-dimensional estimation of quadratic variation based on penalized realized variance

Kim Christensen, Mikkel Slot Nielsen, Mark Podolskij

TL;DR

The paper tackles high-dimensional estimation of quadratic variation for continuous Itô semimartingales by introducing the penalized realized variance (PRV), which imposes a low-rank structure via nuclear-norm regularization and is computed through soft-thresholding the RV eigenvalues. It develops a complete non-asymptotic theory, establishing sharp bounds on estimation error and rank, and proves minimax optimality up to logarithmic factors. A data-driven tuning parameter via subsampling is proposed to facilitate practical implementation, along with a theoretical treatment of local variance estimation. Empirical and simulation results indicate that the PRV effectively identifies a small number of driving factors (typically 1–3) in high dimensions and exhibits rank compression during financial distress, consistent with factor-based asset pricing models.

Abstract

In this paper, we develop a penalized realized variance (PRV) estimator of the quadratic variation (QV) of a high-dimensional continuous Itô semimartingale. We adapt the principle idea of regularization from linear regression to covariance estimation in a continuous-time high-frequency setting. We show that under a nuclear norm penalization, the PRV is computed by soft-thresholding the eigenvalues of realized variance (RV). It therefore encourages sparsity of singular values or, equivalently, low rank of the solution. We prove our estimator is minimax optimal up to a logarithmic factor. We derive a concentration inequality, which reveals that the rank of PRV is -- with a high probability -- the number of non-negligible eigenvalues of the QV. Moreover, we also provide the associated non-asymptotic analysis for the spot variance. We suggest an intuitive data-driven subsampling procedure to select the shrinkage parameter. Our theory is supplemented by a simulation study and an empirical application. The PRV detects about three-five factors in the equity market, with a notable rank decrease during times of distress in financial markets. This is consistent with most standard asset pricing models, where a limited amount of systematic factors driving the cross-section of stock returns are perturbed by idiosyncratic errors, rendering the QV -- and also RV -- of full rank.

High-dimensional estimation of quadratic variation based on penalized realized variance

TL;DR

The paper tackles high-dimensional estimation of quadratic variation for continuous Itô semimartingales by introducing the penalized realized variance (PRV), which imposes a low-rank structure via nuclear-norm regularization and is computed through soft-thresholding the RV eigenvalues. It develops a complete non-asymptotic theory, establishing sharp bounds on estimation error and rank, and proves minimax optimality up to logarithmic factors. A data-driven tuning parameter via subsampling is proposed to facilitate practical implementation, along with a theoretical treatment of local variance estimation. Empirical and simulation results indicate that the PRV effectively identifies a small number of driving factors (typically 1–3) in high dimensions and exhibits rank compression during financial distress, consistent with factor-based asset pricing models.

Abstract

In this paper, we develop a penalized realized variance (PRV) estimator of the quadratic variation (QV) of a high-dimensional continuous Itô semimartingale. We adapt the principle idea of regularization from linear regression to covariance estimation in a continuous-time high-frequency setting. We show that under a nuclear norm penalization, the PRV is computed by soft-thresholding the eigenvalues of realized variance (RV). It therefore encourages sparsity of singular values or, equivalently, low rank of the solution. We prove our estimator is minimax optimal up to a logarithmic factor. We derive a concentration inequality, which reveals that the rank of PRV is -- with a high probability -- the number of non-negligible eigenvalues of the QV. Moreover, we also provide the associated non-asymptotic analysis for the spot variance. We suggest an intuitive data-driven subsampling procedure to select the shrinkage parameter. Our theory is supplemented by a simulation study and an empirical application. The PRV detects about three-five factors in the equity market, with a notable rank decrease during times of distress in financial markets. This is consistent with most standard asset pricing models, where a limited amount of systematic factors driving the cross-section of stock returns are perturbed by idiosyncratic errors, rendering the QV -- and also RV -- of full rank.

Paper Structure

This paper contains 11 sections, 12 theorems, 54 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Theoretical framework
  3. Upper bounds for the PRV
  4. Minimax optimality
  5. Bound on the rank of PRV
  6. Selection of tuning parameter
  7. Estimation of the local variance
  8. Simulation study
  9. Empirical application
  10. Conclusion
  11. Appendix of proofs

Key Result

Proposition 2.1

Let $\widehat{ \Sigma}_{n} = \sum_{k=1}^{d} s_{k} u_{k}u_{k}^{ \top}$ be the orthogonal decomposition of $\widehat{ \Sigma}_{n}$. Then, the unique solution $\widehat{ \Sigma}_{n}^{ \lambda}$ to equation:lasso-estimation is given by

Figures (4)

  • Figure 1: Relative importance of eigenvalues in simulated model.
  • Figure 2: Properties of the PRV.
  • Figure 3: Proportion of variance explained by each eigenvalue.
  • Figure 4: Rank of PRV.

Theorems & Definitions (13)

  • Proposition 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Corollary 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Corollary 2.9
  • Example 2.10
  • ...and 3 more