[Paper Review] ESTIMATION OF INTEGRATED QUADRATIC COVARIATION BETWEEN TWO ASSETS WITH ENDOGENOUS SAMPLING TIMES

Yoann Potiron, Per A. Mykland|arXiv (Cornell University)|Jan 1, 2015

Stochastic processes and financial applications34 references3 citations

TL;DR

This paper proposes a nonparametric Hitting Boundary Time (HBT) model for endogenous sampling in high-frequency covariance estimation between two assets. It establishes a central limit theorem for the Hayashi-Yoshida estimator under this model, derives an asymptotic bias, and provides a bias-corrected estimator and consistent standard error estimator, enabling more accurate inference under realistic sampling mechanisms.

ABSTRACT

When estimating high-frequency covariance (quadratic covariation) of two arbitrary assets observed asynchronously, simple assumptions, such as independence, are usually imposed on the relationship between the prices process and the observation times. In this paper, we introduce a general endogenous two-dimensional nonparametric model. Because an observation is generated whenever an auxiliary process called observation time process hits one of the two boundary processes, it is called the hitting boundary process with time process (HBT) model. We establish a central limit theorem for the Hayashi-Yoshida (HY) estimator under HBT in the case where the price process and the observation price process follow a continuous Ito process. We obtain an asymptotic bias. We provide an estimator of the latter as well as a bias-corrected HY estimator of the high-frequency covariance. In addition, we give a consistent estimator of the associated standard error.

Motivation & Objective

• To address the limitations of existing high-frequency covariance estimators that assume exogenous or independent sampling times.
• To model endogenous observation times where sampling occurs when an auxiliary process hits a boundary, reflecting realistic market microstructure.
• To establish asymptotic theory—specifically a central limit theorem—for the Hayashi-Yoshida estimator under this endogenous HBT framework.
• To derive an explicit expression for the asymptotic bias of the HY estimator under the HBT model.
• To propose a bias-corrected estimator and a consistent estimator of the standard error for practical inference.

Proposed method

• Formalizes a two-dimensional continuous Itô process model where observation times are generated by a hitting boundary process (HBT), with sampling triggered when an auxiliary process hits one of two boundary processes.
• Applies stochastic calculus to derive the asymptotic distribution of the Hayashi-Yoshida estimator under the HBT model, establishing a central limit theorem.
• Derives the asymptotic bias of the HY estimator by analyzing the dependence structure between price processes and observation times under the HBT mechanism.
• Proposes a consistent estimator of the asymptotic bias using realized variation and co-variation measures from the observed data.
• Constructs a bias-corrected version of the HY estimator by subtracting the estimated bias from the original estimator.
• Derives a consistent estimator of the standard error of the bias-corrected estimator, enabling valid inference in finite samples.

Experimental results

Research questions

• RQ1 How does endogenous sampling, induced by a hitting boundary process, affect the asymptotic distribution of the Hayashi-Yoshida estimator for high-frequency covariance?
• RQ2 What is the form of the asymptotic bias in the HY estimator when sampling times are endogenously determined by a boundary-crossing mechanism?
• RQ3 Can a consistent estimator of the asymptotic bias be constructed under the HBT model using observed high-frequency data?
• RQ4 How can the bias-corrected HY estimator be constructed and what are its finite-sample properties?
• RQ5 Is it possible to consistently estimate the standard error of the bias-corrected estimator under the HBT model?

Key findings

• The Hayashi-Yoshida estimator is asymptotically normal under the HBT model, establishing a central limit theorem for endogenously sampled high-frequency data.
• An explicit asymptotic bias is derived for the HY estimator due to the dependence between price processes and observation times under the HBT mechanism.
• A consistent estimator of the asymptotic bias is proposed, which can be computed from observed high-frequency data without requiring knowledge of the underlying boundary processes.
• A bias-corrected version of the HY estimator is constructed by subtracting the estimated bias, improving finite-sample accuracy.
• A consistent estimator of the standard error of the bias-corrected estimator is derived, enabling valid confidence intervals and hypothesis testing.
• The proposed methods are valid under general continuous Itô processes for both price and observation time processes, extending applicability beyond i.i.d. or independent sampling assumptions.