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

Key findings from this study

This research indicates that:

• Integrated ACD models imply infinite-mean durations, requiring new asymptotic theory that addresses the random number of observations within fixed time horizons.
• All five cryptocurrency ETFs examined exhibit infinite-mean inter-trade durations, contradicting the finite-expectation assumptions underlying standard duration models.
• A hypothesis testing framework distinguishes finite-mean from infinite-mean duration specifications, enabling formal assessment of tail behavior in transaction data.

Overview

This research develops asymptotic theory for integrated autoregressive conditional duration (ACD) models, which characterize the time intervals between financial transactions. The study addresses a theoretical gap in understanding ACD processes when integrated specifications imply durations with infinite expectation. The authors establish limit theory for quasi-maximum likelihood estimators under conditions where conventional asymptotic approaches fail due to the random number of duration observations within fixed time periods.

Methods and approach

The authors derive unified asymptotic theory for the quasi-maximum likelihood estimator in integrated ACD frameworks. They develop a hypothesis testing framework to distinguish between finite-mean and infinite-mean duration specifications. The empirical application examines high-frequency trading data from five cryptocurrency exchange traded funds (ETFs), testing whether inter-trade durations exhibit finite expectation.

Results

Analysis of the five cryptocurrency ETFs reveals infinite-mean durations across all cases. The integrated ACD specification is rejected in favor of heavier-tailed alternatives for four of the five ETFs examined. Parameter estimates concentrated near the integrated ACD boundary motivated the formal testing of whether durations possess finite expectation, a commonly implicit assumption in point process models for financial data.

The new testing framework enables researchers to formally assess tail behavior of duration distributions without restricting analysis to conventional parametric specifications. The cryptocurrency trading data demonstrates that realistic market microstructure may violate standard assumptions embedded in existing duration models.

Implications

The theoretical framework extends the econometric toolkit for modeling irregular time series prevalent in financial markets. Rejecting finite-mean assumptions for cryptocurrency ETF trades suggests that models assuming conventional moment conditions may require substantial revision for high-frequency data. The testing methodology enables future empirical research to explicitly evaluate tail assumptions rather than imposing them implicitly.

These findings indicate that duration-based models for financial transactions require careful specification of moment properties. Markets with extreme heterogeneity in inter-trade intervals may exhibit infinite-mean characteristics, necessitating robust inference methods that do not depend on standard moment assumptions. The results have direct implications for risk measurement and market microstructure analysis in cryptocurrency and other digitally-traded asset classes.

Scope and limitations

This summary is based on the study abstract and available metadata. It does not include a full analysis of the complete paper, supplementary materials, or underlying datasets unless explicitly stated. Findings should be interpreted in the context of the original publication.