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

ArXiv ID: 2505.06190 “View on arXiv”

Authors: Giuseppe Cavaliere, Thomas Mikosch, Anders Rahbek, Frederik Vilandt

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.

Keywords: Autoregressive Conditional Duration (ACD), Integrated GARCH, Quasi-Maximum Likelihood Estimator, Asymptotic Theory, Point Process Models, Cryptocurrencies / High-Frequency Trading

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 8.5/10
• Quadrant: Holy Grail
• Why: The paper develops non-standard asymptotic theory for integrated ACD models, involving advanced limit theorems and mixed normal distributions, resulting in a high math complexity score. It is applied to high-frequency cryptocurrency data, with specific findings and testing procedures, indicating substantial empirical implementation, though it focuses on a single dataset.

  flowchart TD
    A["Research Goal: Develop unified asymptotic theory for ACD models<br>and test for finite vs. infinite mean durations"] --> B["Methodology: Quasi-Maximum Likelihood Estimation<br>Unified Asymptotic Theory for Integrated ACD"]
    B --> C["Data: High-Frequency Cryptocurrency ETF Trading Data<br>5 Cryptocurrencies examined"]
    C --> D["Computational Process: Estimation & Hypothesis Testing<br>Testing for infinite expectation using tail index analysis"]
    D --> E{"Key Findings & Outcomes"}
    E --> F["Infinite-Mean Durations Found<br>in all 5 cryptocurrencies"]
    E --> G["Integrated ACD Hypothesis Rejected<br>for 4 out of 5 cryptocurrencies<br>(against alternatives with tail index < 1)"]
    E --> H["New Framework Established<br>for inference on finite/infinite expectation in duration models"]