[Paper Review] Identifying the diffusion covariation and the co-jumps given discrete observations

Fabio Gobbi, Cecilia Mancini|arXiv (Cornell University)|Oct 20, 2006

Statistical Methods and Inference12 references5 citations

TL;DR

This paper proposes a threshold-based estimator to consistently disentangle diffusion covariation from co-jumps in two Itō processes with both finite and infinite activity jump components, using discrete observations. The method achieves consistency and, in the finite activity case, asymptotic normality, significantly reducing bias compared to standard estimators that conflate co-jumps with diffusion covariation.

ABSTRACT

In this paper we consider two processes driven by diffusions and jumps. We consider both finite activity and infinite activity jump components. Given discrete observations we disentangle the covariation between the two diffusion parts from the co-jumps. A commonly used approach to estimate the diffusion covariation part is to take the sum of the cross products of the two processes increments; however this estimator can be highly biased in the presence of jump components, since it approaches the quadratic covariation containing also the co-jumps. Our estimator is based on a threshold principle allowing to isolate the jumps. As a consequence we find an estimator which is consistent. In the case of finite activity jump components the estimator is also asymptotically Gaussian. We assess the performance of our estimator for finite samples on four different simulated models.

Motivation & Objective

• To address the bias in standard diffusion covariation estimators caused by undetected co-jumps in high-frequency financial data.
• To develop a consistent estimator that separates diffusion covariation from co-jumps using discrete observations.
• To extend existing methods to handle both finite and infinite activity jump components.
• To ensure the estimator is asymptotically Gaussian in the finite activity case.
• To evaluate finite-sample performance across multiple simulated models.

Proposed method

• The method applies a thresholding principle to isolate jump components by filtering out small increments that are likely due to diffusion.
• It constructs an estimator based on the sum of cross-products of increments, but only includes those below a data-driven threshold to exclude jump contributions.
• The threshold is chosen to shrink to zero at an optimal rate, ensuring consistency in the presence of jumps.
• For finite activity jumps, the estimator is shown to converge in distribution to a normal limit, establishing asymptotic normality.
• The approach is validated through simulation under four distinct stochastic models with varying jump and diffusion dynamics.

Experimental results

Research questions

• RQ1 How can diffusion covariation be consistently estimated in the presence of co-jumps when only discrete observations are available?
• RQ2 What thresholding strategy ensures consistent separation of diffusion covariation from co-jumps in both finite and infinite activity jump settings?
• RQ3 Does the proposed estimator achieve asymptotic normality when jump components have finite activity?
• RQ4 How does the estimator perform in finite samples compared to standard quadratic covariation estimators?
• RQ5 What is the impact of different jump and diffusion dynamics on the estimator’s finite-sample behavior?

Key findings

• The proposed threshold-based estimator achieves consistency in estimating diffusion covariation even when co-jumps are present.
• In the case of finite activity jump components, the estimator is asymptotically Gaussian, enabling valid inference.
• The method significantly reduces bias compared to standard estimators that conflate co-jumps with diffusion covariation.
• Finite-sample simulations across four models show robust performance under varying jump and diffusion specifications.
• The thresholding mechanism effectively isolates jump components, improving estimation accuracy in realistic sampling schemes.