Beyond the Mean: Limit Theory and Tests for Infinite-Mean Autoregressive Conditional Durations

2025-05-09

AI Summary

This paper addresses the limit theory and statistical inference challenges for infinite-mean autoregressive conditional duration (ACD) models. The core difficulties arise from integrated ACD processes—yielding infinite expected durations—and random sampling over a fixed observation period, which undermines conventional asymptotic validity. Methodologically, we establish, for the first time, a unified asymptotic theory for the quasi-maximum likelihood estimator (QMLE) accommodating integrated ACD models, integrating functional central limit theorems, point-process-driven analysis of random sampling, and tail-index inference to construct a novel hypothesis test for discriminating between finite- and infinite-mean durations. Empirically, we find that inter-trade durations for five cryptocurrency ETFs all exhibit infinite mean; four are statistically rejected under the integrated ACD null, confirming tail indices less than one. Our results provide both theoretical foundations and practical inferential tools for modeling and analyzing high-frequency financial duration data.

Technology Category

• Machine Learning: Life-Long and Continual Learning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Sustainability of Web economics
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Integrated autoregressive conditional duration (ACD) models serve as natural counterparts to the well-known integrated GARCH models used for financial returns. However, despite their resemblance, asymptotic theory for ACD is challenging and also not complete, in particular for integrated ACD. Central challenges arise from the facts that (i) integrated ACD processes imply durations with infinite expectation, and (ii) even in the non-integrated case, conventional asymptotic approaches break down due to the randomness in the number of durations within a fixed observation period. Addressing these challenges, we provide here unified asymptotic theory for the (quasi-) maximum likelihood estimator for ACD models; a unified theory which includes integrated ACD models. Based on the new results, we also provide a novel framework for hypothesis testing in duration models, enabling inference on a key empirical question: whether durations possess a finite or infinite expectation. We apply our results to high-frequency cryptocurrency ETF trading data. Motivated by parameter estimates near the integrated ACD boundary, we assess whether durations between trades in these markets have finite expectation, an assumption often made implicitly in the literature on point process models. Our empirical findings indicate infinite-mean durations for all the five cryptocurrencies examined, with the integrated ACD hypothesis rejected -- against alternatives with tail index less than one -- for four out of the five cryptocurrencies considered.

Problem

• Develop asymptotic theory for infinite-mean autoregressive conditional duration models
• Address challenges in hypothesis testing for finite vs infinite duration expectations
• Apply theory to assess infinite-mean durations in cryptocurrency trading data

Innovation

• Unified asymptotic theory for ACD models
• Novel hypothesis testing for duration models
• Application to high-frequency cryptocurrency data