Extracted main text — title through conclusion, appendix excluded. This is what our citation measures are computed over, published so the extraction can be checked by eye.
69,588 characters · 14 sections · 0 citation commands
{ This paper develops bootstrap methods for likelihood-based inference in autoregressive conditional duration (ACD) models, where the sample size is endogenously determined by durations observed over a fixed time span. This feature fundamentally shapes the asymptotic framework, particularly so when the durations do not have finite expectation. Building on recent limit theory for heavy-tailed and integrated ACD processes, we analyze recursive bootstrap schemes that either fix the time span (yielding a random sample size) or fix the number of durations (yielding a random time span). We establish a bootstrap theory for ACD models that links naturally to renewal theory with random sample sizes. For the fixed-count bootstrap, we prove first-order validity in the finite-mean and boundary cases and characterize the random limiting bootstrap distribution in the infinite-mean case. Although classical bootstrap consistency can fail when the durations have infinite expectation, we argue that the bootstrap remains valid for percentile and reverse-percentile inference and yields asymptotically normal t-statistics. Monte Carlo evidence shows that the proposed methods have good finite sample properties in both finite- and infinite-mean settings, and are robust to distributional misspecification relative to the exponential likelihood. We conclude with an empirical application to cryptocurrency ETFs. }
\noindentKeywords: Autoregressive conditional duration; Bootstrap; Random sample size; Heavy tails; Mixed normality; Point processes.
JEL classification: C22; C32; C52.
Autoregressive conditional duration (ACD) models are designed for irregularly spaced event data. Their central object is the waiting time between consecutive events, for example transaction durations, quote durations, intervention durations or other inter-arrival times in economics and finance. In these applications the econometrician typically observes all events occurring during a given calendar-time window. The number of observations is therefore not chosen directly: it is generated by the stochastic duration process itself.
The ACD model was introduced by Engle and Russell (1998) for durations between financial transactions, and has since become a standard tool for high-frequency durations and other irregularly spaced event data. Important extensions and surveys include Engle (2000), Pacurar (2008), Hautsch (2012), Fernandes, Medeiros and Veiga (2016), Bhogal and Variyam (2019), and Saulo et al. (2025). This literature typically emphasizes the multiplicative conditional-mean structure and its close analogy with autoregressive conditional heteroskedastic (ARCH, GARCH) and multiplicative error model (MEM) specifications. This analogy is useful but, as demonstrated here, it can be misleading for asymptotic inference, and the bootstrap theory. In particular, in ACD models the likelihood is evaluated over a random number of durations observed during a fixed calendar-time span, whereas in ARCH, GARCH and standard MEM models the sample size is treated as deterministic and given ex ante.
To fix ideas, let $\{t_{i}\}_{i\geq 0}$ denote event times, with $ 0=t_{0}<t_{1}<t_{2}<\cdots $, and define the durations by $ x_{i}=t_{i}-t_{i-1}>0$. Over a fixed observation window $[0,T]$, $T>0$, the observed number of events is
The data are the durations $x_{1},\ldots ,x_{n(T)}$. Thus the sample size $ n(T)$ is random and jointly determined with the same durations that enter the likelihood. As mentioned, this differs from the usual ARCH and MEM settings, where the sample size is treated as deterministic.
The randomness of $n(T)$ is crucial. Recent limit theory for ACD models in Cavaliere, Mikosch, Rahbek and Vilandt (2024, 2025, 2026) shows that, under strict stationarity of the duration process $\{x_{i}\}$, the rate and the limiting distribution of likelihood estimators depend on the tail index $ \kappa $ of the stationary distribution of $x_{i}$, which satisfies
for some constant $c_{\kappa }>0$. When $\kappa >1$, durations have finite mean $\mathbb{E}[x_{i}]<\infty $, and the maximum likelihood estimator converges at the calendar-time rate $\sqrt{T}$. At the boundary $\kappa =1$, where $\mathbb{E}[x_{i}]=\infty $, the rate is slower, namely $\sqrt{T/\log T }$. Finally, when $0<\kappa <1$, durations have infinite mean $\mathbb{E} [x_{i}]=\infty $ and the estimator converges at rate $\sqrt{T^{\kappa }}$ and has a mixed normal limiting distribution. These regimes are driven by the large-sample behavior of the random count $n(T)$, which depends on $ \kappa $ being lower, equal, or greater than one.
This paper studies the non-standard consequences of the randomness of the number of durations for bootstrap inference in ACD\ models. Any bootstrap algorithm must decide whether to reproduce the original calendar-time window or the realized number of durations.
A first natural bootstrap scheme preserves calendar time. It recursively generates bootstrap durations $\{x_{i}^{\ast }\}$ until their cumulative sum reaches the original time span $T$, and sets
We label this the random-count bootstrap. It mimics how the original data are collected, but its bootstrap count can be very different from the observed $n(T)$, particularly when $\kappa $ approaches or falls below one.
A second scheme preserves the realized count and sets the number of events in the bootstrap world as $n^{\ast }=n(T)$. We label this the fixed-count bootstrap. The implied bootstrap time span, $T^{\ast }=\sum\nolimits_{i=1}^{n(T)}x_{i}^{\ast }$, need not equal the original $T$, which may seem counterintuitive for point-process data. Nevertheless, it is close to what is often implemented in empirical MEM and ACD applications, it is computationally stable, and as we will argue in the paper it turns out to deliver valid inference without additional conditions other than guaranteeing that the non-bootstrap estimator has a well-defined asymptotic distribution.
The main contribution of the paper is to provide an asymptotic theory for the fixed-count bootstrap. We show that the bootstrap consistently estimates the distribution of the original likelihood estimator when $\kappa \geq 1$. When $0<\kappa <1$, it does not consistently estimate the unconditional mixed-normal distribution, but instead a component thereof. We show that despite this, naive percentile bootstrap intervals remain first-order valid, and the bootstrap $t$ statistic is asymptotically standard normal for all $ \kappa >0$. Moreover, as we document in a simulation study, the bootstrap works well in terms of coverage and average length of the implied confidence intervals, also compared with the random-count bootstrap. This holds for all values of $\kappa $, and also when the innovations of the duration model are not exponentially distributed.
The results in the present paper relate to bootstrap inference for point processes. Cavaliere, Lu, Rahbek and Staerk-Ostergaard (2023) develop bootstrap inference for Hawkes and general point processes under finite-mean conditions. While their parametric recursive bootstrap corresponds to the parametric random count bootstrap (see Section 6 in Cavaliere et al., 2023), their theory does not apply unless $\mathbb{E}[x_{i}]<\infty $. Hence, our problem is different because the ACD recursion can generate durations with infinite expectation. In this case, the counting process grows more slowly than calendar time, and the random sample size induces additional randomness to the asymptotic distribution of the likelihood estimator. This is precisely the feature highlighted in the ACD (non-bootstrap) limit theory of Cavaliere, Mikosch, Rahbek and Vilandt (2024, 2025, 2026). In particular, Cavaliere, Mikosch, Rahbek and Vilandt (2026) show that, for any $\kappa \leq 1$, the randomness of $n(T)$ breaks the conventional `deterministic sample size' asymptotics. The present paper takes the next step by asking which bootstrap schemes remain valid also for infinite-mean durations.
Extant bootstrap literature for the related MEM class is substantial. For instance, Hidalgo and Zaffaroni (2007) study goodness-of-fit testing for ARCH $(\infty )$ models, while Perera, Hidalgo and Silvapulle (2016) develop a bootstrap goodness-of-fit test for a class of ACD models. Perera and Silvapulle (2023) provide bootstrap specification tests for dynamic conditional distribution models. These contributions are directly relevant to implementation, and their algorithms are close in spirit to the fixed-count (residual) bootstrap procedures considered below. However, their asymptotic arguments are formulated for a deterministic number of observations, and they therefore do not address the renewal component generated by (ref), nor the change in rate and limiting law that appears when durations have different tail indexes $\kappa $.
The paper is organized as follows. Section (ref) introduces the model, assumptions and non-bootstrap limit theory. Section (ref) defines the bootstrap schemes and states the bootstrap validity results. Section (ref) presents results from a Monte Carlo study. An empirical illustration based on cryptocurrency ETFs is provided in Section (ref). Section (ref) concludes. The Appendix contains the full proofs and the required auxiliary bootstrap renewal results.
Notation. We write `$\overset{d}{\rightarrow }$', `$ \overset{p}{\rightarrow }$' and `$\overset{a.s.}{\rightarrow }$' for convergence in distribution, probability and almost surely. Conditional bootstrap probability and expectation are denoted by $\mathbb{P}^{\ast }$ and $\mathbb{E}^{\ast }$. Bootstrap convergence is written as `$\overset{ d^{\ast }}{\rightarrow }_{a.s.}$', `$\overset{d^{\ast }}{\rightarrow }_{p}$' or `$\overset{d^{\ast }}{\rightarrow }_{d}$', according to whether the conditional distribution converges almost surely, in probability or weakly as a random probability measure (see Cavaliere and Georgiev, 2020). The notation $\mathcal{MN}$ denotes a mixed-normal distribution.
In this section we summarize the asymptotic (non-bootstrap) limit theory for ACD\ as given in Cavaliere, Mikosch, Rahbek and Vilandt (2024, 2025, 2026). Consider the simple exponential ACD model of order one,
where $\theta =(\omega ,\alpha )^{\prime }$, $\omega >0$, $\alpha >0$, and $ \{\varepsilon _{i}\}$ is i.i.d. exponential with unit mean. The exponential likelihood is the reference likelihood throughout the paper. In the Monte Carlo designs below we allow for non-exponential innovations as a robustness exercise; in that case the same estimator is interpreted as a quasi-maximum likelihood estimator. We let $\theta _{0}=\left( \omega _{0},\alpha _{0}\right) ^{\prime }$ denote the true value of $\theta $, $\omega _{0}>0$, $\alpha _{0}>0$ and make the following assumption throughout:
The MLE\ $\hat{\theta}_{n(T)}=\operatorname{arg\,max}_{\theta \in \Theta } \mathcal{L}_{n(T)}(\theta )$, with log-likelihood function given by
For the scalar null hypothesis $\mathsf{H}_{0}:\alpha =\alpha _{0}$, we also define the studentized statistic
where $\iota _{2}=(0,1)^{\prime }$ and the observed information matrix $ \mathcal{I}_{n(T)}(\hat{\theta}_{n(T)})$ is given by
The stationarity condition $\mathbb{E}[\log (\alpha _{0}\varepsilon _{i})]<0$ in Assumption ((ref)) corresponds, for exponentially distributed innovations, to $\alpha _{0}<\exp (\gamma )\simeq 1.78$, where $\gamma $ is Euler's constant. The tail index $\kappa $ of the stationary solution is the unique positive solution to
Thus $\kappa >1$ if $\alpha _{0}<1$; $\kappa =1$ if $\alpha _{0}=1$; and $ 0<\kappa <1$ if $\alpha _{0}>1$; see Cavaliere, Mikosch, Rahbek and Vilandt (2024). Moreover, let
and define
For later use, the associated counting process theory in Cavaliere et al. (2026) shows that
where $\mu _{0}=\mathbb{E}[x_{i}]=\omega _{0}/(1-\alpha _{0})$, $ c_{0}=\omega _{0}/\mathbb{E}[\varepsilon _{i}\log \varepsilon _{i}]$ and $ \lambda _{\kappa }^{-1/\kappa }$ is a strictly positive $\kappa $-stable random variable.
The asymptotic properties of the MLE\ are presented in the following theorem.
In compact form, the results in Theorem (ref) can be summarized as
with $Z_{\alpha }\sim \mathcal{N}(0,V_{\alpha })$ and $A_{\kappa }=\mu _{0}$ for $\kappa >1$, $A_{\kappa }=c_{0}$ for $\kappa =1$, and $A_{\kappa }=\lambda _{\kappa }^{-1}$ for $0<\kappa <1$, with $\lambda _{\kappa }$ independent of $Z_{\alpha }$ (this notation will be used below).
Theorem (ref) makes clear why bootstrap inference for ACD models is delicate. The convergence rate of the estimator depends on the tail index, while the studentized statistic has a standard normal limit in all regimes. We therefore distinguish between confidence intervals based on studentized and non-studentized statistics. The result also clarifies to what extent the ACD limit theory differs from the more familiar GARCH and MEM asymptotics. For GARCH and MEM likelihoods, the stochastic recurrence generating the data may be similar, but the estimator is normalized by a deterministic sample size. Here the rate is inherited from the renewal limit for $n(T)$. This is why the finite-mean, boundary and infinite-mean cases must be treated separately even though the likelihood contributions have the same formal expression.
A confidence band $\operatorname{CI}_{\tau }$ with nominal coverage $100(1-p)\%$ ( $p\in \left( 0,1\right) $) can be constructed using the studentized $\tau _{n(T)}$ in ((ref)) as
where $z_{p}$ is the $p$th quantile of the standard normal distribution.
Using the results in Theorem (ref) to construct confidence intervals based on the asymptotic distribution of $\hat{\theta}_{n(T)}$ is infeasible in practice due to the presence of nuisance parameters and uncertainty about the true value of $\kappa $. To see this, define the confidence interval $\operatorname{CI}$ as
where $q_{\kappa }(p)$ denotes the $p$ quantile of the asymptotic distribution of $g_{\kappa }(T)^{1/2}(\hat{\alpha}_{n(T)}-\alpha _{0})$. This is clearly infeasible; however, its bootstrap counterpart can easily be constructed, as exemplified in the next section.
A simple model-based bootstrap algorithm can be constructed by generating the duration in the bootstrap world, say $\left\{ x_{i}^{\ast }\right\} $, recursively as
with $x_{0}^{\ast }=x_{0}$ and $n^{\ast }$ the bootstrap sample size. To construct bootstrap confidence intervals it is convenient to refer to the unrestricted bootstrap, which sets $\theta ^{\ast }=\hat{\theta}_{n(T)}$. The bootstrap innovations can be generated either parametrically, setting $ \varepsilon _{i}^{\ast }\sim \operatorname{Exp}(1)$ and i.i.d. conditionally on the data, or non-parametrically, by resampling the scaled residuals
Notice that the scaling enforces
for the bootstrap shocks, while preserving the non-negativity condition $ \varepsilon _{i}^{\ast }>0$ (a.s.).
In the context of the ACD model, the sample size $n^{\ast }$ in the bootstrap world can be defined according to two different schemes. First, the random-count bootstrap keeps the original calendar-time span fixed and sets
Second, the fixed-count bootstrap keeps the realized number of durations fixed and sets
The fixed-count scheme is the focus of the theory below. This choice is close to the practice in the ACD and MEM bootstrap literature, where the observed number of durations is typically held fixed. The main difference is that here $n(T)$ is not a user-chosen deterministic quantity; rather, it is itself a statistic based on the original durations. Consequently, existing proofs of bootstrap validity that treat the sample size in the bootstrap world as deterministic cannot be employed in the current setting; one also needs to show that replacing $n$ by the random count $n(T)$ preserves the relevant likelihood expansions and asymptotic properties. The auxiliary renewal lemmas in the Appendix are used exactly for this purpose.
Define the bootstrap log-likelihood function
where $\psi _{i}^{\ast }(\theta )=\omega +\alpha x_{i-1}^{\ast }$. The bootstrap MLE is
The associated bootstrap observed information and bootstrap $t$ statistic are
A key difference between the mechanics in the fixed-count bootstrap world and the original world is that in the former, the time span covered by the bootstrap durations is random and different from the original time span $ [0,T]$. Precisely, the time span in the bootstrap world is $[0,T^{\ast }]$ with $T^{\ast }:=\sum\nolimits_{i=1}^{n(T)}x_{i}^{\ast }$. The time span $ T^{\ast }$ is not measurable with respect to the original data and in general not expected to be close to $T$, in particular when $\kappa <1$. Indeed, when $\kappa <1$ this bootstrap is unable to replicate the asymptotic distribution of the original estimator, as shown in the following theorem. Despite this, as we argue below, this bootstrap still delivers valid confidence intervals and hypothesis tests.
For $\kappa \geq 1$ the fixed-count bootstrap consistently estimates the limiting distribution of the estimator. For $0<\kappa <1$, the limiting bootstrap measure is not mixed Gaussian as in Theorem (ref); rather, it is random in the limit and reproduces a (Gaussian) component of the limiting mixed normal distribution. As a consequence, classical bootstrap consistency fails in the infinite-mean case. The failure is nevertheless benign for the validity of basic percentile bootstrap inference. To see this, let
denote the normalized original and bootstrap estimators, and let $\hat{F} _{n(T)}^{\ast }(u)=\mathbb{P}^{\ast }(\mathcal{T}_{n(T)}^{\ast }\leq u)$ be the distribution function of $\mathcal{T}_{n(T)}^{\ast }$, conditionally on the data. The one-sided bootstrap p-value is
and we have the following corollary; see also the general bootstrap theory for similar considerations in Cavaliere and Georgiev (2020, Theorem 3.1).
This result ensures validity of bootstrap tests based on $\hat{p} _{n(T)}^{\ast }$ and bootstrap percentile confidence intervals based on $ \hat{F}_{n(T)}^{\ast }$; see, e.g., Remark 3.3 in Cavaliere and Georgiev (2020).
The studentized bootstrap confidence interval equivalent of $\operatorname{CI} _{\tau }$ is given by
where $q_{\tau ^{\ast }}^{\ast }(p)$ is the empirical $p$ quantile of the bootstrap studentized statistics $\tau _{n(T)}^{\ast }$.
The bootstrap equivalent $\operatorname{CI}^{\ast }\,$of the infeasible non-studentized, naive interval $\operatorname{CI}$ is given by
where $q_{\hat{\alpha}^{\ast }}^{\ast }(p)$ is the empirical $p$ quantile of $\hat{\alpha}_{n(T)}^{\ast }$. The formula is equivalently obtained from the quantiles of $g_{\kappa }(T)^{1/2}(\hat{\alpha}_{n(T)}^{\ast }-\hat{\alpha} _{n(T)})$; the factor $g_{\kappa }(T)$ cancels in the endpoint expression.
In terms of testing, if interest is in the scalar null hypothesis $\mathsf{H} _{0}:\alpha =\alpha _{0}$, inference can be based on the bootstrap statistic $\tau _{n(T)}^{\ast }$, see ((ref)). Alternatively, a restricted bootstrap algorithm and test can be employed, where the bootstrap sample is generated recursively as
with $x_{0}^{\star }=x_{0}$ and $\tilde{\theta}_{n(T)}=(\tilde{\omega} _{n(T)},\alpha _{0})^{\prime }$ the restricted estimator of $\theta $, obtained with the null hypothesis imposed. The restricted bootstrap test statistic has the form
where $\hat{\alpha}_{n(T)}^{\star }$ and $\hat{\sigma}(\hat{\alpha} _{n(T)}^{\star })$ are the MLE obtained on the bootstrap sample and its standard error, respectively. Using the arguments in the proof of Theorem (ref), it is straightforward to verify that, under the null hypothesis, $\tau _{n(T)}^{\star }\overset{d^{\star }}{\rightarrow }_{p} \mathcal{N}(0,1)$.
In the case where $\varepsilon _{i}$ are non-exponential with $\mathbb{E} \left[ \varepsilon _{i}\right] =1$ and $\sigma _{\varepsilon }^{2}=\mathbb{V} \left[ \varepsilon _{i}\right] <\infty $, the $\tau _{n(T)}$ statistic is asymptotically $\mathcal{N}\left( 0,\sigma _{\varepsilon }^{2}\right) $, cf. Cavaliere et al. (2026). More specifically, the asymptotic distribution of the estimator is scaled by $\sigma _{\varepsilon }$ for $\kappa =1$ and $ \kappa >1$ (we conjecture that the same result holds when $\kappa <1$ as well), and standard inference becomes invalid. While standard robustification of the $\tau _{n(T)}$ statistic is well known and can in principle be employed, an advantage of the non-parametric bootstrap proposed in this paper is that it does not require the underlying assumption of exponential innovations. That is, the bootstrap, as also shown in the simulation exercise of the next section, remains valid, regardless of the distribution of $\varepsilon _{i}$.
In this section we provide a Monte Carlo study to evaluate the finite-sample properties of the proposed fixed-count bootstrap. We also aim at providing some comparisons with the random-count bootstrap, where the number of observations is random in the bootstrap world. For all designs, we consider $ M=10000$ Monte Carlo replications and $B=399$ bootstrap replications.
The baseline data-generating process $\left\{ x_{i}\right\} _{i=1}^{n(T)}$ is (ref)-(ref) with $\omega _{0}=1$, and where, for a given target duration tail index $\kappa $, the value of $\alpha _{0}$ is chosen from
Specifically, for a given observation window $[0,T]$, $T>0$, durations $ \{x_{i}\}_{i=1}^{n(T)}$ are simulated using a `burn-in' sample of $d=1000$ observations,
with $\varepsilon _{i}$ i.i.d. and initial value $x_{-d}=0$. To examine robustness to different innovation tails, in addition to $\varepsilon _{i}$ being exponentially distributed, we consider $\varepsilon _{i}$ being Lomax (Pareto Type II) distributed. The Lomax distribution function, under the constraint $\mathbb{E}\left[ \varepsilon _{i}\right] =1$, is given by
with the exponential distribution obtained as the limiting case by letting $ s\rightarrow \infty $. This distribution has a right power-law tail with tail index $s>1$,
Thus, $\sigma _{\varepsilon }^{2}(s)=\mathbb{V}\left[ \varepsilon _{i}\right] <\infty $ provided $s>2$, in which case $\sigma _{\varepsilon }^{2}(s)= \mathbb{E}[\varepsilon _{i}^{2}]-1=s/(s-2)$.
In the simulations, we vary the number of finite moments for $\varepsilon _{i}$ by considering shape parameters as given by $s\in \{2.1,3,\infty \}$, where $\sigma _{\varepsilon }^{2}\left( 2.1\right) =21$, $\sigma _{\varepsilon }^{2}\left( 3\right) =3$, and $\sigma _{\varepsilon }^{2}\left( \infty \right) =1$. In the non-exponential case, the values of $ \alpha _{0}$ that correspond to particular target values of $\kappa \in \{0.5,1.0,1.1\}$ are obtained as the unique solution to ((ref)) which for any $\kappa <s$ is given by
with $\Gamma \left( \cdot \right) $ denoting the Gamma function. Note that for the exponential case this reduces to (ref). Table (ref) reports the corresponding values of $\alpha _{0}$ used in the simulations.
The observation-period span $T$ is calibrated so that the median event count across replications, $\operatorname{med}n\left( T\right) $, belongs to $ \{200,400,800,1600\}$. For each Monte Carlo replication the unrestricted estimator $\hat{\theta}_{n(T)}$ is computed, and next $B$ bootstrap samples are generated from the random-count and the fixed-count bootstrap algorithms, respectively. The reported confidence-interval results focus on $ \alpha$ and compare the intervals introduced in Sections (ref) and (ref): the random-count and fixed-count versions of the basic bootstrap interval $\operatorname{CI}^{\ast }$, the asymptotic studentized interval $\operatorname{CI}_{\tau }$, and the random-count and fixed-count versions of the bootstrap-$t$ interval $\operatorname{CI}_{\tau }^{\ast }$.
We also consider tests of the scalar null hypothesis $\mathsf{H}_{0}:\alpha =\alpha _{0}$. The non-bootstrap test uses
and compares $|\tau |$ with the standard normal critical value $q=1.96$. The bootstrap tests are based on a restricted bootstrap generated under the null hypothesis. Specifically, the bootstrap durations are generated with $\alpha ^{\ast }=\alpha _{0}$ and $\omega ^{\ast }=\tilde{\omega}_{n(T)}$, where $ \tilde{\omega}_{n(T)}$ is the restricted estimator computed from the original data. The bootstrap innovations are resampled from the scaled restricted residuals. For the fixed-count version we set $n^{\ast }=n(T)$; for the random-count version we use the recursively generated count $n^{\ast }(T)$ defined in (ref). The bootstrap statistic is
where the bootstrap standard error is computed from the bootstrap observed information as
with $\iota _{2}=(0,1)^{\prime }$, as in (ref).
The main quantities of interest are empirical coverage probabilities, average interval lengths, and empirical rejection probabilities. The theory predicts that fixed-count intervals should remain valid even when $0<\kappa _{0}<1$, while avoiding the additional variation in $n^{\ast }(T)$ that is built into the random-count bootstrap.
Tables (ref)--(ref) report empirical coverage probabilities (CPs) and average lengths (ALs) of confidence intervals (CIs).
For exponential innovations, $s=\infty $, the asymptotic studentized interval $\operatorname{CI} _{\tau }$ has coverage close to the nominal level. The fixed-count bootstrap intervals also perform well. The bootstrap-$t$ interval $\operatorname{CI} _{\tau }^{\ast }$ has coverage around $0.92$--$0.93$ across the three values of $\kappa _{0}$, while the basic fixed-count interval $\operatorname{CI} ^{\ast }$ is similarly close to nominal coverage. The random-count basic interval $\operatorname{CI} _{\ast }^{\ast }$ often gives coverage close to nominal as well, but at the cost of noticeably longer average lengths.
The advantages of the residual bootstrap are more visible when the innovations are non-exponential. For $s=3$, the standard asymptotic interval under-covers substantially, with CPs typically around $0.76$--$0.82$ even for the larger samples. In contrast, the fixed-count bootstrap-$t$ intervals have CPs close to $0.91$--$0.93$, and the basic fixed-count intervals have coverage close to the nominal level. For $s=2.1$, where the variance of the innovations is finite but large, the asymptotic interval performs poorly, whereas the bootstrap intervals continue to give a substantial correction. This supports the main practical point of Section (ref): the residual bootstrap can provide useful robustness to misspecification of the exponential likelihood.
Table (ref) reports empirical rejection probabilities for the test of $\mathsf{H}_{0}:\alpha =\alpha _{0}$. When $s=\infty $, the normal critical value gives rejection frequencies close to the nominal level. For non-exponential innovations, however, the same test over-rejects markedly, with rejection frequencies increasing as the innovation variance increases. The restricted bootstrap tests correct this size distortion. Both the fixed-count and random-count versions deliver rejection frequencies close to the nominal level across the reported values of $\kappa _{0}$, $s$ and $\operatorname{med}n(T)$. The fixed-count version is therefore competitive with the random-count version, while being simpler and avoiding the additional randomness in the bootstrap event count.
We illustrate the bootstrap procedures using intra-day trade durations for five exchange-traded funds (ETFs) tracking cryptocurrency prices. The data are the same as in Cavaliere, Mikosch, Rahbek and Vilandt (2026). The ETFs are the Grayscale Bitcoin Mini Trust (ticker: BTC), Grayscale Ethereum Mini Trust (ETH), Grayscale Bitcoin Trust (GBTC), Grayscale Ethereum Trust (ETHE) and Bitwise Bitcoin (BITB). The sample covers 35 trading days starting on January 2, 2025. During regular trading hours (9:30am to 4:00pm EST), this gives a raw calendar span $T=35\cdot 23400=819000$ seconds.
Durations are measured in seconds and are obtained from NASDAQ limit order book data through the LOBSTER database (https://data.lobsterdata.com/ ). The raw intra-day durations are adjusted for deterministic intra-day patterns using cubic splines with knots every 30 minutes, as is standard in empirical duration analysis; see Hautsch (2012, Ch.3).\footnote{ The processed data can be downloaded at https://github.com/CMRV-ACD/IACD.} In the bootstrap implementation below, the calendar span used by the random-count bootstrap is the sum of the adjusted durations, $T=\sum_{i=1}^{n(T)}x_i$. The resulting numbers of durations are 19366 for BTC, 35492 for ETH, 157620 for GBTC, 120104 for ETHE and 51917 for BITB.
The empirical specification is the standard ACD(1,1) model
This is the empirically standard extension of the first-order autoregressive duration model studied in the theory above. We apply the same fixed-count and random-count residual bootstrap logic to this specification and report inference for $\omega $, $\alpha $, $\beta $ and the persistence parameter $ \alpha +\beta $. As in the simulations, the fixed-count bootstrap sets $ n^{\ast }=n(T)$, while the random-count bootstrap keeps fixed the span of adjusted durations, $T=\sum_{i=1}^{n(T)}x_{i}$.
Table (ref) reports unrestricted QML estimates and confidence intervals. The estimates of $\alpha +\beta $ are above one for all five ETFs, with values between 1.002 for BITB and 1.018 for ETH. The fixed-count bootstrap intervals for $\alpha +\beta $ are tight and lie above one for BTC, ETH, GBTC and ETHE. For BITB, the fixed-count bootstrap intervals include one, indicating that the evidence against the integrated boundary is weaker for this series. The random-count basic intervals are substantially wider, especially for the intercept, reflecting the additional variation induced by the random bootstrap event count. For $\omega$, lower endpoints of confidence intervals are truncated at zero whenever the corresponding untruncated lower endpoint is negative.
Table (ref) reports bootstrap tests of the integrated ACD hypothesis $\alpha +\beta =1$. The table gives the observed $t$ statistic and the $2.5\%$ and $97.5\%$ bootstrap quantiles obtained under the restriction $\alpha ^{\ast }+\beta ^{\ast }=1$. We report quantiles based both on unrestricted residuals and on restricted residuals. The latter are used to generate the restricted bootstrap draws and lead to slightly sharper critical values, but the empirical conclusions are unchanged.
Using the restricted-residual fixed-count bootstrap, the integrated ACD hypothesis is rejected for BTC, ETH, GBTC and ETHE. For BITB, the observed statistic $3.11$ lies within the bootstrap quantile interval $[-4.66,4.02]$, and the integrated specification is therefore not rejected at the 5 percent level. Since all unrestricted estimates satisfy $\hat{\alpha}+\hat{\beta} \geq 1$, we also do not reject the null hypothesis of infinite expected durations against the finite-mean alternative $\alpha +\beta <1$. Overall, the empirical illustration points to very persistent, heavy-tailed duration dynamics in cryptocurrency ETF trading, and it also shows that bootstrap critical values can matter empirically relative to the standard Gaussian approximation.
This paper studies bootstrap inference in ACD models when the number of observed durations is random. The treatment of the number of events in the bootstrap world is central. A random-count bootstrap preserves the calendar-time span, while a fixed-count bootstrap preserves the realized number of durations. The latter is simple to implement and closely aligned with existing empirical practice, but its validity cannot be justified by standard deterministic-sample-size bootstrap arguments.
The fixed-count bootstrap is valid in the finite-mean and boundary regimes, $ \kappa \geq 1$. In the infinite-mean regime, $0<\kappa <1$, the bootstrap limiting measure is random and classical consistency fails. Nevertheless, the random limiting measure matches the conditional Gaussian component of the asymptotic mixed normal distribution, which is sufficient for first-order validity of the basic bootstrap interval and for standard normality of the bootstrap $t$ statistic. This provides a practical inference route that does not require direct estimation of stable-law quantiles.
The main takeaway and implication of this analysis is that bootstrap inference for duration models should not be justified only by analogy between the ACD model and GARCH or MEM models. While this analogy is useful for constructing the likelihood function and related statistics, it does not by itself justify the validity of bootstrap inference. Existing bootstrap approaches for MEM, as in Perera, Hidalgo and Silvapulle (2023), and point-process bootstrap methods, as in Cavaliere, Lu, Rahbek and Staerk-Ostergaard (2023), provide important building blocks. The main contribution of this paper is to show how these ideas can be combined with the recent ACD limit theory of Cavaliere, Mikosch, Rahbek and Vilandt (2026) to provide bootstrap confidence intervals that remain meaningful and first-order valid across finite- and infinite-mean duration regimes.
The authors gratefully acknowledge support from the Independent Research Fund Denmark (DFF Grant 7015-00028) and the Italian Ministry of University and Research (PRIN 2020 Grant 2020B2AKFW).
Bhogal, S.K., and Variyam, R.T. (2019), Conditional duration models for high-frequency data: a review on recent developments. Journal of Economic Surveys, 33, 252--273.
Cavaliere, G., and Georgiev, I. (2020), Inference under random limit bootstrap measures. Econometrica, 88, 2547--2574.
\noindentCavaliere, G., Lu, Y., Rahbek, A., and Staerk-Ostergaard, J. (2023), Bootstrap inference for Hawkes and general point processes. Journal of Econometrics, 235, 133--165.
\noindentCavaliere, G., Mikosch, T., Rahbek, A., and Vilandt, F. (2024), Tail behavior of ACD models and consequences for likelihood-based estimation. Journal of Econometrics, 238, Article 105613.
\noindentCavaliere, G., Mikosch, T., Rahbek, A., and Vilandt, F. (2025), A comment on: “Autoregressive conditional duration: a new model for irregularly spaced transaction data”. Econometrica, 93, 719--729.
Cavaliere, G., Mikosch, T., Rahbek, A., and Vilandt, F. (2026), Beyond the mean: limit theory and tests for infinite-mean autoregressive conditional durations. Journal of the Royal Statistical Society Series B: Statistical Methodology, qkag053.
\noindentCavaliere, G., Nielsen, H.B., and Rahbek, A. (2017), On the consistency of bootstrap testing for a parameter on the boundary of the parameter space. Journal of Time Series Analysis, 38, 513--534.
Engle, R.F. (2000), The econometrics of ultra-high-frequency data. Econometrica, 68, 1--22.
\noindentEngle, R.F., and Russell, J.R. (1998), Autoregressive conditional duration: a new model for irregularly spaced transaction data. Econometrica, 66, 1127--1162.
Fernandes, M., Medeiros, M.C., and Veiga, A. (2016), The (semi-)parametric functional coefficient autoregressive conditional duration model. Econometric Reviews, 35, 1221--1250.
Gut, A. (2009), Stopped Random Walks: Limit Theorems and Applications, 2nd ed. Springer.
\noindentGut, A. (2013), Probability: A Graduate Course. Springer.
Hautsch, N. (2012), Econometrics of Financial High-Frequency Data. Springer.
\noindentHidalgo, J., and Zaffaroni, P. (2007), A goodness-of-fit test for ARCH$(\infty)$ models. Journal of Econometrics , 141, 835--875.
Kristensen, D., and Rahbek, A. (2005), Asymptotics of the QMLE for a class of ARCH(q) models. Econometric Theory, 21, 946--961.
\noindentLee, S.-W., and Hansen, B.E. (1994), Asymptotic theory for the GARCH(1,1) quasi-maximum likelihood estimator. Econometric Theory, 10, 29--52.
\noindentPacurar, M. (2008), Autoregressive conditional duration models in finance: a survey of the theoretical and empirical literature. Journal of Economic Surveys, 22, 711--751.
Perera, I., Hidalgo, J., and Silvapulle, M.J. (2016), A goodness-of-fit test for a class of autoregressive conditional duration models. Econometric Reviews, 35, 1111--1141.
\noindentPerera, I., and Silvapulle, M.J. (2023), Bootstrap specification tests for dynamic conditional distribution models. Journal of Econometrics, 235, 949--971.
Saulo, H., Suvra, P., Rubens, S., Vila, R., and Dasilva, A. (2025), Parametric quantile autoregressive conditional duration models with application to intraday value-at-risk forecasting. Journal of Forecasting, 44, 589--605.