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

TL;DR

This paper advances the asymptotic theory for autoregressive conditional duration models by incorporating the integrated ACD case, where durations can have infinite mean. It establishes that the (quasi-)MLE converges at the non-standard rate √(t/ln t) to a Gaussian limit when α + β = 1, and provides a unified framework for estimation and inference that covers both finite- and infinite-mean scenarios. The authors develop a testing suite for the integrated ACD restriction and for infinite mean, with theoretical results supported by Monte Carlo evidence. An empirical application to high-frequency cryptocurrency ETF trading data reveals heavy-tailed durations with infinite mean in most assets, underscoring the importance of the new non-standard inference tools in duration modeling and market microstructure analysis.

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.

Figures (6)

• Figure 1: QQ-plot. Quantiles of the $\tau_{t}$ statistic in (\ref{['eq def t stat']}) against the $N(0,1)$-distribution. For each $\alpha _{0}\in\{0.15,0.5,0.85\}$, values of $t$ are such that median number of observations, $\operatorname*{med}\left\{ n\left( t\right) \right\} \in$$\left\{ 100,500,2500,12500,62500\right\}$. Simulations based on $\varepsilon_{i}$ exponentially distributed. Number of Monte Carlo-replications $M=10000.$
• Figure 2: Rejection frequencies under alternative. Case of $\alpha _{0}=0.85$. Rejection frequencies for $\tau_{t}>q_{R}$ (right hand side of $c=0$) and $\tau_{t}<q_{L}$(left hand side), with $q_{R}$ [$q_{L}$] the size-adjusted $0.95$ [$0.05$] quantiles. Solid line: median number of observations $\operatorname*{med}\{n(t)\}=62500$ for $c=0$; dashed, dotted-dashed and dotted lines: $\operatorname*{med}\{n(t)\}$ equal to $12500$, $2500$ and $500$, respectively. Number of Monte Carlo-replications $M=10000.$
• Figure 3: Rejection frequencies under alternative. Case of $\alpha _{0}=0.5$. Rejection frequencies for $\tau_{t}>q_{R}$ (right hand side of $c=0$) and $\tau_{t}<q_{L}$(left hand side), with $q_{R}$ [$q_{L}$] the size-adjusted $0.95$ [$0.05$] quantiles. Solid line: median number of observations $\operatorname*{med}\{n(t)\}=62500$ for $c=0$; dashed, dotted-dashed and dotted lines: $\operatorname*{med}\{n(t)\}$ equal to $12500$, $2500$ and $500$, respectively. Number of Monte Carlo-replications $M=10000.$
• Figure 4: Rejection frequencies under alternative. Case of $\alpha _{0}=0.15$. Rejection frequencies for $\tau_{t}>q_{R}$ (right hand side of $c=0$) and $\tau_{t}<q_{L}$(left hand side), with $q_{R}$ [$q_{L}$] the size-adjusted $0.95$ [$0.05$] quantiles. Solid line: median number of observations $\operatorname*{med}\{n(t)\}=62500$ for $c=0$; dashed, dotted-dashed and dotted lines: $\operatorname*{med}\{n(t)\}$ equal to $12500$, $2500$ and $500$, respectively. Number of Monte Carlo-replications $M=10000.$
• Figure 5: Durations and Diurnal Pattern. Right column: Diurnally adjusted durations $x_{i}$ (in seconds) as a function of calender time. Left column: Estimated diurnal intra-daily pattern in durations $x_{i}$ as a function of time (corresponding to 9:30am-4pm).