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.

翻译：本文提出了一种分析点过程间超前-滞后关系的新理论框架，特别关注其在高频金融数据中的应用。我们重点研究两个订单到达时间戳序列之间的超前-滞后关系。Dobrev和Schaumburg的开创性工作基于时间戳的交叉计数提出了跨市场交易活动的非模型化度量方法。尽管该方法被证实能产生可靠结果，但其原始公式本质上依赖于离散时间观测，这一局限性在本研究中得到解决。具体而言，我们将两个点过程的超前-滞后关系估计问题，转化为估计二元平稳点过程的交叉配对相关函数（CPCF）形态的问题——该量在神经科学和空间统计学文献中已有深入研究。在此框架下，主导的超前-滞后时间被定义为CPCF最尖锐峰值的位点。基于此解释，Dobrev和Schaumburg跨市场活动度量中的峰值位点可视为上述意义上的超前-滞后时间估计量。我们进一步提出基于核密度估计的替代性超前-滞后时间估计量，证明其具备理想的理论性质并提供更优的数值性能。高频金融数据的实证证据验证了所提方法的有效性。