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We establish new results for estimation and inference in financial durations models, where events are observed over a given time span, such as a trading day, or a week. For the classical autoregressive conditional duration (ACD) models by Engle and Russell (1998, Econometrica 66, 1127--1162), we show that the large sample behavior of likelihood estimators is highly sensitive to the tail behavior of the financial durations. In particular, even under stationarity, asymptotic normality breaks down for tail indices smaller than one or, equivalently, when the clustering behaviour of the observed events is such that the unconditional distribution of the durations has no finite mean. Instead, we find that estimators are mixed Gaussian and have non-standard rates of convergence. The results are based on exploiting the crucial fact that for duration data the number of observations within any given time span is random. Our results apply to general econometric models where the number of observed events is random.
\noindentKeywords:\ Financial durations; autoregressive conditional duration (ACD); tail index; quasi maximum likelihood; mixed normality.
In the seminal papers by Engle and Russell (1998) and Engle (2000), autoregressive conditional duration (ACD) models were introduced for modeling durations, or waiting times, between financial events, and to analyze liquidity in financial markets. Financial events are observed over a given period of time, such as a (trading) day, a week, or a year; hence, both the size and the number of durations are random variables. As we demonstrate, the randomness of the number of events has a major impact on asymptotics and inference in dynamic duration models. Moreover, as detailed below, existing results cover alone the case where the number of events is non-random and therefore are not applicable to estimation of ACD models over a given time span. In this paper, we provide the missing asymptotic analysis for likelihood-based estimators. We specifically show that the randomness of the number of events plays a crucial role and leads to a new distributional theory (at non-standard rates of convergence)\ for likelihood estimators and related test statistics. The derivation of these novel results requires non-standard asymptotic arguments, combining new results on the tail behaviour of the durations with renewal theory.
The ACD models are by now quite popular in financial econometrics; see e.g. Hautsch (2012) and Fernandes, Medeiros and Veiga (2016) for applications and theory in the context of high-frequency data and Pacurar (2008) for a general survey. Applications of dynamic duration models such as the ACD\ are extensively used also in different areas of economics; see e.g. Hamilton and Jord\`{a} (2002) or Aquilina, Budish and O'Neill (2022).
Let $[0,T]$ denote the observation period, where we observe $n$ event times $\left\{ t_{i}\right\} _{i=1}^{n}$, $0<t_{1}<t_{2}<\cdots<t_{n}\leq T$, with corresponding durations $x_{i}=t_{i}-t_{i-1}$, $i=1,...,n$, $t_{0}=0$. As noted in Engle and Russell (1998), $n$ is the realization at time $t=T$ of the stochastic counting process $N_{t}$, $t\geq0$, given by
In particular, the number of events $N_{T}$, in the observation period $[0,T]$, is as mentioned a random variable.
The most known dynamic duration model is the ACD\ of Engle and Russell (1998) which in its simplest version (ACD\ of order one) is given by
where $\theta=(\omega,\alpha)^{\prime}$ and $\psi_{i}(\theta)$ is the conditional (duration) rate of the ith waiting time $x_{i}$, i.e., conditional on $\mathcal{F}_{i-1}=\sigma(x_{i-1},x_{i-2},\ldots)$. The innovations $\{\varepsilon_{i}\}$ are assumed i.i.d., strictly positive, with unit mean, $\mathbb{E}[\varepsilon_{i}]=1$. If $\varepsilon_{i}$ is exponentially distributed this is referred to as exponential ACD (EACD).
With parameters $\theta=(\omega,\alpha)^{\prime}$, for $\omega>0$, $\alpha \geq0$, and observation period $[0,T]$, the EACD log-likelihood function is {given by}
Then $\hat{\theta}_{T}=\arg\max_{\theta}L_{T}\left( \theta\right) $ denotes the maximum likelihood estimator (MLE) of $\theta$ in the case of i.i.d. exponentially distributed $\{\varepsilon_{i}\}$, otherwise we refer to it as a quasi maximum likelihood estimator (QMLE).
Engle and Russell (1998) note that the log-likelihood function in ((ref)) has the same form as the log-likelihood function for the autoregressive conditional heteroskedastic (ARCH) model with Gaussian innovations, and quote standard asymptotic theory from ARCH models in Lee and Hansen (1996); see also Fernandes and Grummig (2006), Hautsch\ (2012, Theorem 5.2), Allen, Felix, McAleer and Peiris (2008) and Sin (2014) for a similar approach to inference. Importantly, this approach treats $N_{T}$ as deterministic; that is, sampling is by number of durations and not over a fixed, predetermined observation period $[0,T]$. Importantly, the results for deterministic $N_{T}$ cannot be applied to the case of random $N_{T}$, as analyzed here.
To give an idea of the difference in arguments between the two different sampling schemes, a key insight is that the fact that the number of observations $N_{T}$ is random implies that classical laws of large numbers (LLNs) and central limit theorems (CLTs) are no longer directly applicable to likelihood-related quantities such as score and information. For instance, it is known from renewal process theory (see e.g. Gut, 2009) that $N_{T}$ $\rightarrow\infty$ is not sufficient for the LLN\ or the CLT to apply to series of the form $Y_{T}=\sum_{i=1}^{N_{T}}\xi_{i}$, where both $N_{T}$ and the random variables $\{\xi_{i}\}$ are defined in terms of the durations $\{x_{i}\}$; such series appear repeatedly in the asymptotic theory for ACD. In contrast to the deterministic $N_{T}$ case, the large sample behaviour of $Y_{T}$ is intimately related to the large sample properties of the counts $N_{T}$, which, again, depends on the tail properties (and existence of moments) of the marginal distribution of the stationary and ergodic duration $x_{i}$. Such dependence leads to a novel asymptotic theory, based on non-standard arguments.
Specifically, as we show in this paper, the asymptotic theory for the MLE\ crucially depends on the tail behavior and existence of moments for the ergodic and stationary durations $\{x_{i}\}$, with the tail behaviour characterized by the tail index $\kappa>0$ of the marginal distribution of $x_{i}$; $P(x_{i}>z)\sim c_{\kappa}z^{-\kappa}$ as $z\rightarrow\infty$ for some constant $c_{\kappa}>0$. We show that while asymptotic normality holds for $\kappa>1$,\thinspace or equivalently, when the durations have finite mean, asymptotic normality breaks down for $\kappa<1$. This is a crucial fact, given that a wide range of tail indices is witnessed in empirical applications on duration data. Thus, for example, Hill estimation of $\kappa$ yields $\hat{\kappa}=2.1>2$ for the IBM\ transaction data analyzed in Engle and Russell (1998) and $\hat{\kappa}=2.5>2$ on the durations between tweets in Cavaliere, Lu, Rahbek and St\ae rk-\O stergaard (2022). Moreover, $\hat {\kappa}=1.4\in\left( 1,2\right) $ for the DJIA\ data from Embrechts, Liniger and Lin (2011), while $\hat{\kappa}=0.7<1$ on SPY\ transaction data over a single trading day (2019:07:31). Notably, while asymptotic normality holds for $\kappa>1$, the Gaussian finite sample approximation is poor for the case of infinite variance $\kappa<2$ and indeed invalid for the case of infinite mean where $\kappa<1$.
A preview of our results is as follows. In classic settings, with $\{x_{i}\}$ i.i.d. with finite mean, the number of events per unit of time $N_{T}/T$ converges in probability to a strictly positive constant, in which case LLNs and CLTs for $\sum_{i=1}^{N_{T}}\xi_{i}$ can usually be verified; see e.g. Gut (2009) for a survey. In the ACD\ setting, whether this holds depends on the tail index $\kappa$. On the one hand, if $\kappa>1$, hence $\mathbb{E}[x_{i}]=\mu\in(0,\infty)$, and $N_{T}/T$ is such that $N_{T}/T=1/\mu+o(1)$ a.s. However, even in this simpler case, existing (renewal) theory does not include stationary and ergodic $x_{i}$, and we provide the needed extensions to the theory here. On the other hand, if $\kappa<1$, hence $\mathbb{E}[x_{i}]=\infty$, then $N_{T}/T$ converges (a.s.)\ to zero as $T\rightarrow\infty$ and neither the classic LLN nor the CLT apply to $\sum_{i=1}^{N_{T}}\xi_{i}$. New tools are required for the asymptotic theory and, in particular, we establish the novel result that $N_{T}/T^{\kappa}$ converges in distribution to a random variable with an unfamiliar distribution, and for which we provide an explicit expression in terms of a $\kappa$-stable random variable.
These convergence results for $N_{T}$ are essential for establishing the asymptotic distribution of the QMLE. Specifically, we show that, provided $\mu=\mathbb{E}[x_{i}]<\infty$, $\hat{\theta}_{T}-\theta_{0}$ (with $\theta_{0}$ denoting the true value) is indeed asymptotically Gaussian when normalized by the standard deterministic $\sqrt{T}$-rate. However, while $\mathbb{E}[x_{i}]<\infty$ is indeed sufficient for $\sqrt{T}$-convergence to the Gaussian distribution, the quality of the Gaussian approximation in finite samples is demonstrated to be very poor when $\mathbb{E}[x_{i}^{2}]=\infty$, or $\kappa<2$, and deteriorating as the tail index $\kappa$ approaches one. Hence even when $\mathbb{E}[x_{i}]<\infty$ these results question the usefulness of the $\sqrt{T}$-Gaussian approximation for likelihood estimators in ACD models. In the case $\kappa<1$, the fact that $N_{T}/T^{\kappa}$ converges in distribution -- and not in probability -- to a non-standard random variable implies that the derivation of the limiting distribution of $\hat{\theta}_{T}-\theta_{0}$ is non-standard. In particular we show that the information is random in the limit, and that this results in a limiting mixed Gaussian distribution of $\hat{\theta}_{T}-\theta_{0}$ with a convergence rate which depends on the value of tail index $\kappa<1$. A further, novel result that follows from our results is that the $t$ ratio for (univariate)\ hypotheses on $\theta$ is asymptotically normal provided $\kappa>1$ or $\kappa<1$. The local power function of the test, however, crucially depends on $\kappa$. The case $\kappa=1$ is not covered by our theorem, and hence particular attention should be paid to applications where estimated parameters are close to the boundary case $Ex_{i}=\infty$.
To sum up, our results show that, in contrast to ARCH\ models where the marginal distribution of the data does not play any role in the asymptotic theory, for ACD\ models this is indeed crucial, as the tail index of the duration determines the speed of convergence of the estimators as well as their asymptotic distribution. As already mentioned this is of empirical relevance, as both the case of infinite and finite mean durations ($\kappa<1$ and $\kappa>1$, respectively) are found in applications. Moreover, our findings are not specific to ACD models, but apply to general econometric method where the number of observations over a given time span needs being treated as a random process; see also Section (ref).
The paper is structured as follows. In Section (ref) we discuss the tail behavior of ACD processes, and provide new results for the related counting process $N_{T},$ $T\geq0$. In Section (ref) we present the main asymptotic theory. A discussion about the implications for inference and some concluding remarks are given in Section (ref). All proofs are provided in the Appendix. In the following, `$\overset{p}{\rightarrow}$', `$\overset{\mathrm{a.s.}} {\rightarrow}$', and `$\overset{d}{\rightarrow}$' refer to convergence in probability, almost surely and in distribution, respectively, in all cases when $T\rightarrow\infty$. A generic element of a strictly stationary sequence $\{y_{i}\}$ is denoted by $y$.
In this section we derive the required results on the tail properties of the durations and on the asymptotic behaviour of the random number of durations $N_{T}$. Results of this kind are neither present nor required in the classical ARCH case, where $N_{T}$ is deterministic.
We consider the sequence\ $x_{i}=\psi_{i}\varepsilon_{i}$, $i\in{\mathbb{Z}}$, given as the solution to the ACD equation (ref), and state explicit conditions for stationarity and geometric ergodicity of $\{x_{i}\}$ as well as for power-law tails of $x$ with index $\kappa$. The range of the values $\kappa$ will be crucial for our asymptotic\ theory.
The results are initially stated for general positive i.i.d. distributed innovations $\{\varepsilon_{i}\}$.
The results in Lemma (ref) complement existing results on ARCH\ processes; see e.g. Embrechts, Kl\"{u}ppelberg and Mikosch (1997), Buraczewski, Damek and Mikosch (2016), and allow in particular one to assess the existence of moments of ACD\ processes. Thus, we find that for $\alpha<1$ the mean is finite, $\mathbb{E}[x]<\infty$, while the variance $\mathbb{V}[x]$ is finite for $\alpha$ in the smaller region $(0,1/s)$, where $s^{2} =\mathbb{E}[\varepsilon^{2}]$. For $1<\alpha<a_{u}$, while $\{x_{i}\}$ is a strictly stationary and geometrically ergodic sequence, only fractional moments (of order less than one) of $x$ are finite.
Next, we consider the benchmark model where $\varepsilon$ is exponentially distributed (EACD).
In particular, we observe the surprisingly simple explicit relationship between $\alpha$ and $\kappa=\kappa\left( \alpha\right) $ in (ref) which comes from the properties of the exponential distribution. {Such a simple relationship does not exist for general distributions of $\varepsilon$ and more general functional forms of $\psi_{i} $.}
In Lemma (ref) below we collect some novel asymptotic\ results for the counting process $N_{T}$, $T\geq0$, which are needed for the asymptotic\ analysis of the QMLE of the ACD process.
Our results are general and of independent interest, in particular as the dependence of the durations sequence\ is an uncommon condition in the literature on renewal theory; there it is typically assumed that the durations are i.i.d. or at most $m$-dependent (e.g. finite moving average); see e.g. Gut (2009), Janson (1983). Moreover, and also new with respect to existing theory, we present results for the convergence of the counting process $N_{T}$ when durations have a tail index $\kappa<1$.
Recall initially that $N_{T}$ is defined in terms of the dependent sequence $\left\{ x_{i}\right\} $, cf. (ref), with $x_{i}$ defined in ((ref)). As in Lemma (ref) we consider here the general case of positive i.i.d. innovations $\left\{ \varepsilon_{i}\right\} $ with unit mean. The following result provides convergence\ rates for $N_{T}$ as $T\rightarrow\infty$.
It is worth noticing that for all cases (i)--(iv), $N_{T}\rightarrow\infty$ a.s. as a consequence\ of the ergodic theorem. However, the convergence\ rates are quite distinct, depending on $\kappa$. Thus, $N_{T}/T\rightarrow1/\mu$ a.s. for $\kappa>1$, while, for $\kappa<1$, $N_{T}/T^{\kappa}$ converges in distribution\ to the positive random variable $\lambda_{\kappa}$, and in particular, $N_{T}/T\overset{p}{\rightarrow}0$. For $\kappa>2$, $N_{T} /T-1/\mu$ satisfies the CLT\ with standard $\sqrt{T}$-rate, while for $1<\kappa<2$, the rate $T^{(\kappa-1)/\kappa}$ gets slower as $\kappa$ gets closer to 1. We also note that the $\kappa$-stable limiting random variable $\eta_{\kappa}$ has power-law tail{ with index $\kappa$.} Importantly, for the novel result on the distributional convergence of $N_{T}/T^{\kappa}$ for $\kappa<1$, the limiting variable $\lambda_{\kappa}$ has exponentially decaying tails; cf. Theorem 2.5.2 in Zolotarev (1986).
In this section we derive the asymptotic properties of the (Q)MLE\ $\hat{\theta}_{T}=\arg\max_{\theta}L_{T}\left( \theta\right) $, with $L_{T}\left( \theta\right) $ defined in ((ref)). Note that, as is common practice, $L_{T}\left( \theta\right) $ is defined without the additional term corresponding to the fact that no events are observed in the end-period $(t_{N_{T}},T]$. We show in Appendix (ref) that this term has no influence on the asymptotic results.
We start in Section (ref) by discussing the behavior of the score and information, which is key to the asymptotic analysis. Then, in Section (ref), we present the main results on the asymptotic behavior of $\hat{\theta}_{T}$.
With the likelihood function $L_{T}(\theta)$ as given in ((ref)), the corresponding score and information functions, evaluated at the true value $\theta=\theta_{0}$, are given by
where $\psi_{i}=\psi_{i}(\theta_{0})$. In what follows, we always assume that the conditions of Lemma (ref) are satisfied. In particular, (ref) has a strictly stationary geometrically{ ergodic} solution $\{x_{i}\}$ with tail index $\kappa>0$.
Consider first the case $\kappa>1$, where we have the following result on the large sample behavior of $S_{T}$ and $I_{T}$ at standard rates of convergence.
Next turn to the case $\kappa<1$ such that $\mathbb{E}[x]=\infty$. As shown in the next, the score and information converge at slower rates than usual. More specifically, turning to the information, it follows by Lemma (ref) that $N_{T}/T^{\kappa}\overset{d}{\rightarrow }\lambda_{\kappa}$ and (see the proof of Lemma (ref) below) $N_{T}^{-1}I_{T}\overset{\mathrm{a.s.}}{\rightarrow}\Omega$. Hence,
That is, the rate of convergence is indeed slower than standard when $\kappa<1$, and the observed information is random in the limit due to the random variable $\lambda_{\kappa}$. Similarly, non-standard convergence rates as a function of $\kappa$ also apply to the score as we state the following lemma for the EACD.
We are now in the position to state the asymptotic distribution of the QMLE $\hat{\theta}_{T}$ of $\theta$. As for the score, the limit behavior of the QMLE\ depends on the tail behavior of the durations $\left\{ x_{i}\right\} $. As mentioned, the influence of the right power-law tail of $x$ on the QMLE is in contrast to QMLE theory for ARCH and GARCH processes where the shape of the unconditional distribution does not matter. We show here that for ACD\ processes the power-law tails determine the limiting distribution\ of the QMLE $\hat{\theta}_{T}$ as well as the rate of convergence. This result appears surprising, given that, apart from the random summation index, the ACD (log-)likelihood function is identical to the ARCH Gaussian likelihood function.
Specifically, while $\sqrt{T}$-asymptotic normality holds when the tail index $\kappa$ is above one, when $\kappa<1$, the speed of convergence and the limiting distribution are non-standard. In particular, for the case $\kappa>1$ the following result holds.
Theorem (ref), which is based on combining classic likelihood expansions with the results for a random summation index $N_{T}$ in Section (ref), shows that asymptotic normality at the $\sqrt{T}$-rate holds even if the durations have infinite variance, $\mathbb{E}[x^{2}]=\infty $. However, it can be shown that the quality of the asymptotic approximation deteriorates as the tail index $\kappa=\kappa(\alpha_{0})$ approaches one. This reflects the fact that the asymptotic\ results for the QMLE when $\kappa>1$ are essentially derived by replacing the random indices $N_{T}$ in the likelihood function\ (and its derivatives)\ by the deterministic function\ $T/\mu$;\ this replacement, however, happens with a much larger error term for $\kappa\in\left( 1,2\right) $ than in the finite variance case ($\kappa>2$), due to slow convergence\ rates of $N_{T}/T-1/\mu$ (cf. Lemma (ref)) and the widespread limit distribution.
As an explanation to the fact that while the rate of convergence is standard $\sqrt{T}$ for all $\kappa>1$, the convergence to the Gaussian limit for $\kappa\in(1,2)$ slows down when compared to the (finite variance) case $\kappa>2$, consider here the score $S_{T}$. By {Lemma (ref) (iii),} \[ T^{-1/2}S_{T}=[T^{\left( 1-\kappa\right) /\kappa}\hat{\gamma}_{\kappa }/(2{\sqrt{\mu})}+1/\sqrt{\mu}+o_{p}\left( 1\right) ]\hat{Z}_{T}, \] where $\hat{\gamma}_{\kappa}=T^{\left( \kappa-1\right) /\kappa}\left( N_{T}/T-1/\mu\right) \rightarrow_{d}\gamma_{\kappa}$ (a $\kappa$-stable random variable) and $\hat{Z}_{T}=N_{T}^{-1/2}S_{T}\rightarrow_{d}\left( \tau\Omega\right) ^{1/2}\,\boldsymbol{Z}$. Additionally, $\gamma_{\kappa}$ is non-standard distributed with {a power-law tail with index $\kappa$, and is more widespread as }$\kappa$ diminishes. Thus, as $\kappa$ approaches one, $T^{\left( 1-\kappa\right) /\kappa}\hat{\gamma}_{\kappa}/(2{\sqrt{\mu})}$ converges to zero at a slower speed, and the convergence (in distribution) of $T^{-1/2}S_{T}$ to the Gaussian distribution slows down{. This is in contrast} to the case $\kappa>2$, where by Lemma (ref) (ii), $T^{-1/2}S_{T}=[T^{-1/2}\hat{\eta_{T}}/{(2\sqrt{\mu})}+1/\sqrt{\mu} +o_{p}\left( 1\right) ]\hat{Z}_{T}$, with $\hat{\eta_{T}}=T^{1/2} (N_{T}/T-1/\mu)$ asymptotically Gaussian. In particular, the rate is independent of $\kappa$ in this case.
We illustrate this in Figure (ref), where we report Q-Q plots of $T^{1/2}(\hat{\alpha}_{T}-\alpha_{0})$ against the Gaussian distribution when the data follows an EACD process with $\mathbb{E}[x]=1$, for different values of $T$ and $\kappa$. The figure clearly shows how the tail index of the durations influences the quality of the Gaussian approximation in finite time intervals. It can also be seen that as $\kappa$ gets closer to one, the asymptotic approximation requires larger values of $T$ to be accurate.
For $\kappa<1$, as previously emphasized, $\hat{\theta}_{T}-\theta_{0}$ is not asymptotically Gaussian distributed.
Thus for $\kappa<1$, the estimators are asymptotically mixed Gaussian; moreover the rate of convergence $T^{\kappa/2}$ is lower than the standard $T^{1/2}$ rate and depends on the value of $\kappa$. The non-Gaussianity in ((ref)) is clearly illustrated in the upper panel of Figure (ref), which reports Q-Q plots of $T^{\kappa/2}(\hat{\alpha}_{T}-\alpha_{0})$ against a zero-mean Gaussian distribution for $\kappa=0.5$ and different values of $T$ ($\omega_{0}$ is selected such that the median of $x_{i}$ is about one).
In the previous section we have shown that, for the case of a finite mean of the durations, the (Q)MLE\ is indeed asymptotically normal at the standard $\sqrt{T}$-rate while, for the case of infinite mean, the limiting distribution is a mixture and convergence attains at a lower rate.
In terms of inference, from the Theorems (ref) and (ref) we can derive the following new result, which shows that $t$-ratios (or quasi likelihood ratio statistics) are asymptotically standard Gaussian ($\chi^{2}$) distributed, irrespectively of the tail index of the durations $\kappa$ being above or below unity. That is, while $\kappa$ affects the distributional theory for of (Q)MLE, asymptotic inference based on $t$ tests (or likelihood ratio tests)\ is standard, and asymptotic validity holds irrespective of the tail index of the marginal distribution of the durations $\{x_{i}\}$.
Convergence of the $t$ ratios to the Normal distribution is illustrated in the lower panel of Figure (ref), where Q-Q plots of $t_{n}$ against the $N\!\!\left( 0,1\right) $ distribution are reported for increasing sample sizes and for $\kappa<1$. The figure clearly shows that extremely large observation periods are required for the normal asymptotic approximation to be accurate. This implies that in empirical applications, and differently from inference in ARCH models, inspection of the tails of the marginal distribution of the data is a key step to be taken prior to any empirical analysis.
Finally, we note that, in terms of theory, the result in Corollary (ref) for $\kappa<1$ is similar to the mixed Gaussian limit results, as employed e.g. in the cointegration analysis of non-stationary variables; see Johansen (1991) and Phillips (1991).
Our new results demonstrate the sensitivity of the limiting distribution of the QMLE\ in ACD\ models to the tail behaviour, or equivalently finiteness of moments, of the durations. This clearly contrasts the previous asymptotic results which, by treating the number of durations as deterministic and hence referring to ARCH\ asymptotic theory, does not depend on the finiteness of moments, nor on the tail behaviour. Stated differently, sampling over a fixed period of time (hence implying a random number of events) leads to new non-standard theory, while sampling over a fixed number of events (hence implying a random length of observation period) leads to standard\ theory from ARCH models.
All results can be generalized to more general ACD\ models, in particular to the much applied ACD model where $\psi_{i}=\omega+\alpha x_{i-1}+\beta \psi_{i-1}$, that is, the ACD\ analogue of the GARCH(1,1), as well as its extensions. We have refrained from doing so here to keep the presentation simple, and thereby focus on the main new insights.
Finally, it is worth noticing that our findings and arguments, are not specific to the models for time series of durations. Indeed, they apply to any econometric method where the number of observations needs being treated as random. For example, asymptotic theory for daily realized volatility, see Li, Mykland, Renault, Zhang and Zheng (2013), treats summations such as $\sum_{i=1}^{N_{T}}(p_{t_{i}}-p_{t_{i-1}})^{2}$, where $p_{t}$ is the (log-)price at time $t$; since the number of trades within a day, $N_{T}$, is random, our results could be applied to cases where $x_{i}=t_{i}-t_{i-1}$ have heavy tails.
We are grateful to Federico Bandi, Marcelo Fernandes, Nikolaus Hautsch and Marcelo Medeiros for comments and discussions. We also thank participants at the SoFiE 2022 conference (U Cambridge), the Aarhus Workshop in Econometrics (Aarhus U), the 3rd High Voltage Econometrics meeting and the Bologna/Rome-Waseda Time Series Workshop. We also thank Roberto Ren\`{o} for providing the SPY duration data. A. Rahbek and G. Cavaliere gratefully acknowledge support from the Danish Council for Independent Research (DSF Grant 015-00028B). Part of G. Cavaliere's research was supported by the Italian Ministry of University and Research (PRIN 2020 Grant 2020B2AKFW). Thomas Mikosch's research is partially supported by Danmarks Frie Forskningsfond Grant No 9040-00086B.
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