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Pre-averaging estimators of the ex-post covariance matrix in noisy diffusion models with non-synchronous data

Kim Christensen, Silja Kinnebrock, Mark Podolskij

Abstract

We show how pre-averaging can be applied to the problem of measuring the ex-post covariance of financial asset returns under microstructure noise and non-synchronous trading. A pre-averaged realised covariance is proposed, and we present an asymptotic theory for this new estimator, which can be configured to possess an optimal convergence rate or to ensure positive semi-definite covariance matrix estimates. We also derive a noise-robust Hayashi-Yoshida estimator that can be implemented on the original data without prior alignment of prices. We uncover the finite sample properties of our estimators with simulations and illustrate their practical use on high-frequency equity data.

Pre-averaging estimators of the ex-post covariance matrix in noisy diffusion models with non-synchronous data

Abstract

We show how pre-averaging can be applied to the problem of measuring the ex-post covariance of financial asset returns under microstructure noise and non-synchronous trading. A pre-averaged realised covariance is proposed, and we present an asymptotic theory for this new estimator, which can be configured to possess an optimal convergence rate or to ensure positive semi-definite covariance matrix estimates. We also derive a noise-robust Hayashi-Yoshida estimator that can be implemented on the original data without prior alignment of prices. We uncover the finite sample properties of our estimators with simulations and illustrate their practical use on high-frequency equity data.
Paper Structure (22 sections, 8 theorems, 113 equations, 1 figure, 3 tables)

This paper contains 22 sections, 8 theorems, 113 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Theoretical setup
  3. Microstructure noise
  4. Pre-averaging of high-frequency data
  5. Modulated realised covariance
  6. Asymptotic properties of MRC
  7. Consistency
  8. The central limit theorem
  9. Choosing $\theta$ in practice
  10. Positive semi-definite estimators
  11. Mapping MRC into a realised kernel estimator
  12. Applying the MRC to non-synchronous data
  13. A pre-averaged Hayashi-Yoshida estimator
  14. An estimator of the asymptotic covariance matrix
  15. Asymptotic theory for covariance, regression and correlation
  16. ...and 7 more sections

Key Result

Theorem 1

Assume that $\mathbb{E} \left( | \epsilon^{j} |^{4} \right) < \infty$ for all $j = 1,...,d$ and $(k_{n}, \theta)$ satisfy Eq. kthetarelation. As $n \to \infty$, it holds that

Figures (1)

  • Figure 1: MRC-based beta.

Theorems & Definitions (8)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Proposition 1
  • Theorem 6
  • Lemma 1