On lead-lag estimation of non-synchronously observed point processes

TL;DR

This paper develops a new theoretical framework for estimating lead-lag relationships between point processes, especially in high-frequency financial data, addressing limitations of previous discrete-time based methods.

Contribution

It formulates lead-lag estimation as the shape of the cross-pair correlation function and proposes a kernel density estimator with improved theoretical and numerical performance.

Findings

The proposed estimator outperforms existing methods in simulations.

Empirical results confirm the effectiveness of the new approach on financial data.

The framework links lead-lag estimation to well-studied correlation functions.

Abstract

This paper introduces a new theoretical framework for analyzing lead-lag relationships between point processes, with a special focus on applications to high-frequency financial data. In particular, we are interested in lead-lag relationships between two sequences of order arrival timestamps. The seminal work of Dobrev and Schaumburg proposed model-free measures of cross-market trading activity based on cross-counts of timestamps. While their method is known to yield reliable results, it faces limitations because its original formulation inherently relies on discrete-time observations, an issue we address in this study. Specifically, we formulate the problem of estimating lead-lag relationships in two point processes as that of estimating the shape of the cross-pair correlation function (CPCF) of a bivariate stationary point process, a quantity well-studied in the neuroscience and spatial statistics literature. Within this framework, the prevailing lead-lag time is defined as the location of the CPCF's sharpest peak. Under this interpretation, the peak location in Dobrev and Schaumburg's cross-market activity measure can be viewed as an estimator of the lead-lag time in the aforementioned sense. We further propose an alternative lead-lag time estimator based on kernel density estimation and show that it possesses desirable theoretical properties and delivers superior numerical performance. Empirical evidence from high-frequency financial data demonstrates the effectiveness of our proposed method.