Beyond the mean: limit theory and tests for infinite-mean autoregressive conditional durations

Abstract

Integrated autoregressive conditional duration (ACD) models serve as counterparts to integrated generalized autoregressive conditional heteroskedastic models used for financial returns. However, despite their resemblance, asymptotic theory for ACD is still incomplete. Central challenges arise from the facts that (i) integrated ACD processes imply durations with infinite expectation and (ii) conventional asymptotic approaches break down due to the randomness in the number of durations within a fixed observation period. We fill this gap in the literature and provide a unified asymptotic theory for the (quasi)maximum likelihood estimator for integrated ACD models. Based on the new theoretical 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 exchange traded fund (ETF) trading data. Motivated by parameter estimates near the integrated ACD region, 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 five cryptocurrency ETFs examined, and we reject the integrated ACD hypothesis in favour of heavier-tailed alternatives for four out of the five ETFs.