A Note on Asynchronous Challenges: Unveiling Formulaic Bias and Data Loss in the Hayashi-Yoshida Estimator

TL;DR

This paper investigates the intrinsic bias and data loss in the Hayashi-Yoshida estimator caused by its telescoping property, formalizes the conditions for this bias, and proposes algorithms to quantify and mitigate data loss in asynchronous data scenarios.

Contribution

It formalizes the formulaic bias in the HY-estimator, quantifies data loss due to this bias, and introduces algorithms to compute and minimize data loss in asynchronous observations.

Findings

Equal rates of Poisson processes minimize data loss at 25%.

The paper provides necessary and sufficient conditions for the bias to occur.

An algorithm to compute nonextant data points is proposed.

Abstract

The Hayashi-Yoshida (\HY)-estimator exhibits an intrinsic, telescoping property that leads to an often overlooked computational bias, which we denote,formulaic or intrinsic bias. This formulaic bias results in data loss by cancelling out potentially relevant data points, the nonextant data points. This paper attempts to formalize and quantify the data loss arising from this bias. In particular, we highlight the existence of nonextant data points via a concrete example, and prove necessary and sufficient conditions for the telescoping property to induce this type of formulaic bias.Since this type of bias is nonexistent when inputs, i.e., observation times, $\Pi^{(1)} :=(t_i^{(1)})_{i=0,1,\ldots}$ and $\Pi^{(2)} :=(t_j^{(2)})_{j=0,1,\ldots}$, are synchronous, we introduce the (a,b)-asynchronous adversary. This adversary generates inputs $\Pi^{(1)}$ and $\Pi^{(2)}$ according to two independent homogenous Poisson processes with rates a>0 and b>0, respectively. We address the foundational questions regarding cumulative minimal (or least) average data point loss, and determine the values for a and b. We prove that for equal rates a=b, the minimal average cumulative data loss over both inputs is attained and amounts to 25\%. We present an algorithm, which is based on our theorem, for computing the exact number of nonextant data points given inputs $\Pi^{(1)}$ and $\Pi^{(2)}$, and suggest alternative methods. Finally, we use simulated data to empirically compare the (cumulative) average data loss of the (\HY)-estimator.