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

TL;DR

This paper develops a unified asymptotic theory for integrated autoregressive conditional duration (ACD) models, addressing challenges posed by infinite-mean durations and randomness in event counts, and applies it to high-frequency cryptocurrency data.

Contribution

It introduces a comprehensive asymptotic framework for ACD models, including integrated cases, and proposes new hypothesis tests for finite versus infinite duration expectations.

Findings

Durations in cryptocurrency markets have infinite mean.

The integrated ACD hypothesis is rejected in most cases.

New asymptotic theory enables inference on duration tail behavior.

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.