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\title{ }
\begin{center}
{\LARGE \textsc{asymptotics for the generalized autoregressive conditional
duration model}}
\footnote{
\hspace{-7.2mm}
$^{a}
$Department of Economics, University of Bologna, Italy and Department of Economics, University of Exeter, UK.
\newline$^{b}
$Department of Mathematical Sciences, University of Copenhagen, Denmark.
\newline$^{c}$Department of Economics, University of Copenhagen, Denmark.
\newline
We thank Matias Cattaneo and seminar participants at Princeton University for comments and suggestions.
A.~Rahbek and G.~Cavaliere gratefully acknowledge support from the Independent Research Fund Denmark (DFF Grant 7015-00028) and
the Italian Ministry of University and Research (PRIN 2020 Grant 2020B2AKFW). T.~Mikosch's research is partially supported by the Independent Research Fund Denmark (DFF Grant 9040-00086B).
Correspondence to: Giuseppe Cavaliere, Department of
Economics, University of Bologna, email [email removed].
}
\addtocounter{footnote}{-1}
{\normalsize \vspace{0.1cm} }
{\large \textsc{Giuseppe Cavaliere}}$^{a}${\large \textsc{, Thomas Mikosch}
}$^{b}$, {\large \textsc{Anders Rahbek}}$^{c}$
{\large \textsc{and Frederik Vilandt}}$^{c}${\normalsize \vspace{0.2cm}
\vspace{0.2cm}}
May 25, 2023{\normalsize \vspace{0.2cm}\vspace{0.2cm}}
Abstract\vspace{-0.15cm}
\end{center}
Engle and Russell (1998, \emph{Econometrica}, 66:1127--1162) apply results
from the GARCH\ literature to prove consistency and asymptotic normality of
the (exponential) QMLE\ for the generalized autoregressive conditional
duration (ACD)\ model, the so-called ACD(1,1), under the assumption of strict
stationarity and ergodicity. The GARCH results, however, do not account for
the fact that the number of durations over a given observation period is
random. Thus, in contrast with Engle and Russell (1998), we show that strict
stationarity and ergodicity alone are not sufficient for consistency and
asymptotic normality, and provide additional sufficient conditions to account
for the random number of durations. In particular, we argue that the durations
need to satisfy the stronger requirement that they have finite mean. \bigskip
\medskip\noindent\textsc{Keywords}:\ autoregressive conditional duration
(ACD); quasi maximum likelihood.
\section{Introduction}
\label{sec intro}\textsc{In the seminal paper }by Engle and Russell (1998, ER
henceforth), autoregressive conditional duration (ACD) models were introduced
for modeling durations between financial transactions. Given some observation
period $[0,T]$, say, with $n(T)$ observed event times $\left\{ t_{i}\right\}
_{i=1}^{n(T)}$, $0<t_{1}<t_{2}<\cdots<t_{n(T)}\leq T$, the durations $x_{i}$
are given by $x_{i}=t_{i}-t_{i-1}$ and modeled as
\begin{align}
x_{i} & =\psi_{i}\left( \theta\right) \varepsilon_{i},\text{
\ \ }i=1,\ldots,n(T)\,,\label{eq ACD_1}\\
\psi_{i}\left( \theta\right) & =\omega+\alpha x_{i-1}+\beta\psi
_{i-1}\left( \theta\right) \label{eq ACD_2}
\end{align}
where the innovations $\{\varepsilon_{i}\}$ are i.i.d., strictly positive,
with unit mean, $\mathbb{E}[\varepsilon_{i}]=1$. The quasi\ maximum likelihood
estimator (QMLE) of $\theta=(\omega,\alpha,\beta)^{\prime}\in\Theta
\subset\mathbb{R}^{3}$ is defined as $\hat{\theta}_{T}=\arg\max_{\theta
\in\Theta}L_{n(T)}\left( \theta\right) $, with $L_{n(T)}\left(
\theta\right) $ the exponential likelihood,
\begin{equation}
L_{n(T)}\left( \theta\right) =-\sum_{i=1}^{n(T)}\ell_{i}(\theta)\text{,
}\ell_{i}(\theta)=\log\psi_{i}\left( \theta\right) +\frac{x_{i}}{\psi
_{i}\left( \theta\right) }\,,\qquad T\geq0\,, \label{eq: EACD likelihood}
\end{equation}
with $\omega,\alpha$ and $\beta$ positive, and initial values $(x_{0},\psi
_{0}(\theta))^{\prime}=\gamma$.
ER\ note that the likelihood function in (\ref{eq: EACD likelihood}) is
identical to the likelihood function of the GARCH(1,1) model with Gaussian
innovations. Hence, for their main result (p. 1135), ER\ refer to Lee and
Hansen (1994) to conclude that under strict stationarity and ergodicity of the
durations $x_{i}$, $\hat{\theta}_{T}$ is consistent and asymptotically normal
at the usual rate; importantly, their conditions allow for $\alpha+\beta\geq
1$, and hence for durations with no finite mean. In contrast, using a new
lemma which extends the arguments in Lee and Hansen (1994) to the ACD case, we
argue that the additional condition $\alpha+\beta<1$, which implies a finite
mean, is sufficient.
To establish asymptotic normality, standard arguments require that the
(normalized) score and information, evaluated at the true value $\theta
=\theta_{0}$:
\begin{align}
S_{n(T)} & =\left. \tfrac{\partial L_{n(T)}(\theta)}{\partial\theta
}\right\vert _{\theta=\theta_{0}}=n(T)^{-1/2}\sum_{i=1}^{n(T)}\xi_{i}
\,,\quad\xi_{i}=\left. \tfrac{\partial\ell_{i}(\theta)}{\partial\theta
}\right\vert _{\theta=\theta_{0}}\label{eq S}\\
I_{n(T)} & =\left. -\tfrac{\partial^{2}L_{n(T)}(\theta)}{\partial
\theta\partial\theta^{\prime}}\right\vert _{\theta=\theta_{0}}=n(T)^{-1}
\sum_{i=1}^{n(T)}\zeta_{i}\,,\qquad\zeta_{i}=\left. -\tfrac{\partial^{2}
\ell_{i}(\theta)}{\partial\theta\partial\theta^{\prime}}\right\vert
_{\theta=\theta_{0}} \label{eq I}
\end{align}
satisfy a central limit theorem (CLT)\ and a law of large numbers (LLN),
respectively. The ACD setting, however, is not standard as the number of
observations $n(T)$ is \emph{random} and not independent of the sequences
$\{\xi_{i}\}$ and $\{\zeta_{i}\}$. Note in this respect that the fact that the
CLT\ and the LLN\ hold for the case of a deterministic number $n$ of
observations, that is
\begin{align}
S_{n} & =n^{-1/2}\sum_{i=1}^{n}\xi_{i}\overset{d}{\rightarrow}N\left(
0,\Omega_{S}\right) \text{,}\,\label{eq CLT for deterministic n}\\
I_{n} & =n^{-1}\sum_{i=1}^{n}\zeta_{i}\,\overset{\text{a.s.}}{\rightarrow
}\Omega_{I}\text{,} \label{eq LLN for deterministic n}
\end{align}
does not imply that their random $n(T)$ analogues in\textbf{ }(\ref{eq S}
)-(\ref{eq I}) hold. Therefore, arguments based on Lee and Hansen (1994) as in
ER, which assume $n$ deterministic, do not apply directly.
In this note, by using a new Lemma which extends Lee and Hansen (1994)\ to the
case of a random number of observations, we show that under the additional
more restrictive\textbf{ }condition $\alpha_{0}+\beta_{0}<1$ (or,
equivalently, that the durations have finite unconditional expectation) the
QMLE\ is consistent and asymptotically normal at the standard $\sqrt{T}$ rate.
The key additional condition needed is that the random number of durations
$n(T)$ satisfies
\begin{equation}
n(T)/T\overset{\text{a.s.}}{\rightarrow}1/\mu\text{ , with }\mu=\mathbb{E}
\left( x_{i}\right) \text{.} \label{eq N_T/T convergence}
\end{equation}
This in turn (as $n(T)\rightarrow\infty$ a.s.) is sufficient for the
deterministic $n$ LLN in (\ref{eq LLN for deterministic n}) to imply that its
random $n$ analogue (\ref{eq I}) holds. To establish the random $n$ CLT
in\textbf{ }(\ref{eq S}), we replace the deterministic $n$ CLT\ in
(\ref{eq CLT for deterministic n}) with its stronger functional version
\[
S_{n}\left( \cdot\right) =n^{-1/2}\sum_{i=1}^{\left\lfloor n\cdot
\right\rfloor }\xi_{i}\overset{d}{\rightarrow}\Omega_{S}^{1/2}B\left(
\cdot\right) \text{,}\,
\]
where $B$ is a standard multivariate Brownian motion.
We finally notice that when $\alpha_{0}+\beta_{0}>1$, which implies ergodicity
provided also $E[\ln\left( \alpha_{0}\varepsilon_{i}+\beta_{0}\right) ]<0$
holds, asymptotically normality is no longer guaranteed; results in Cavaliere,
Mikosch, Rahbek and Vilandt (2022) for the simple ACD\ model with $\beta=0$,
suggest that $\sqrt{T}$ asymptotic normality indeed breaks down in this case.
\section{Main result}
In order to derive the asymptotic distribution of the QMLE\ for the ACD model
given by (\ref{eq ACD_1})-(\ref{eq ACD_2}) we first introduce the following
general lemma, which extends the results in Lee and Hansen (1994) to allow for
a random number of observations.
\begin{lemma}
\label{new: main lemma}Consider $Q_{n}(\varphi)\in\mathbb{R}$, which is a
random function of the deterministic sample size $n$ and the parameter
$\varphi\in\Phi\subseteq\mathbb{R}^{k}$. Assume that $Q_{n}(\cdot
):\mathbb{R}^{k}\rightarrow\mathbb{R}$ is three times continuously
differentiable\ in $\varphi$, and that for $\varphi_{0}$ in the interior of
$\Phi$ it holds that as $n\rightarrow\infty$:$\smallskip$
\noindent$
\begin{tabular}
[c]{ll}
(C.1) & $n^{-1/2}\partial Q_{[n\cdot]}(\varphi_{0})/\partial\varphi\overset
{w}{\rightarrow}\Omega_{S}^{1/2}B(\cdot)$, $\ \Omega_{S}>0$,\\
(C.2) & $-n^{-1}\partial^{2}Q_{n}(\varphi_{0})/\partial\varphi\partial
\varphi^{\prime}\overset{\text{a.s}.}{\rightarrow}\Omega_{I}>0$,\\
(C.3) & $\max_{h,i,j=1,...,k}\sup_{\varphi\in N(\varphi_{0})}\left\vert
n^{-1}\frac{\partial^{3}Q_{n}(\varphi)}{\partial\varphi_{h}\partial\varphi
_{i}\partial\varphi_{j}}\right\vert \leq\tau_{n}\rightarrow\tau$ a.s.,
\end{tabular}
\ \smallskip$
\noindent where $B\left( \cdot\right) $ is a $k$-dimensional Brownian
motion, $N(\varphi_{0})$ is a neighborhood of $\varphi_{0},$ and
$0<\tau<\infty$. Moreover, with $n\left( t\right) $, $t\geq0$, a counting
process defined on the same probability space as $Q_{n}\left( \varphi\right)
$, assume that with $c\in(0,\infty)$ a constant:$\smallskip$
\noindent$
\begin{tabular}
[c]{lc}
(C.4) & As $T\rightarrow\infty$, $n\left( T\right) /T\overset{\text{a.s.}
}{\rightarrow}c$.
\end{tabular}
\ \smallskip$
\noindent Consider next $Q_{n\left( T\right) }\left( \varphi\right) $
which is a random function of the random sample size $n\left( T\right) $ and
$\varphi\in\Phi\subseteq\mathbb{R}^{k}$. Then, as $T\rightarrow\infty$, with
probability tending to one, there exists a fixed open neighborhood
$U(\varphi_{0})\subseteq N(\varphi_{0}),$ $\varphi_{0}\in U(\varphi_{0})$,
such that:$\smallskip$
\noindent$
\begin{tabular}
[c]{ll}
(i) & There exists a maximum point $\hat{\varphi}_{T}$ of $Q_{n\left(
T\right) }(\varphi)$ in $U(\varphi_{0})$ and $Q_{n(T)}(\varphi)$ is concave
in\\
& $U(\varphi_{0})$; in particular, $\hat{\varphi}_{T}$ is unique and solves
$\partial Q_{n\left( T\right) }(\hat{\varphi}_{T})/\partial\varphi=0$,\\
(ii) & $\hat{\varphi}_{T}\overset{p}{\rightarrow}\varphi_{0}$,\\
(iii) & $T^{1/2}(\hat{\varphi}_{T}-\varphi_{0})\overset{d}{\rightarrow
}N(0,\Sigma)$, $\Sigma=c\Omega_{I}^{-1}\Omega_{S}\Omega_{I}^{-1}$.
\end{tabular}
\ $\bigskip
\end{lemma}
The proof of Lemma \ref{new: main lemma} is given in the appendix. Note that
Assumption (C.4) can be replaced by $n\left( T\right) \rightarrow\infty$
a.s. and $n\left( T\right) /T\overset{p}{\rightarrow}c$,\thinspace
\thinspace$0<c<\infty\,$.
Our main result is as follows.
\begin{theorem}
\label{thm main}For the ACD\ model (\ref{eq ACD_1})-(\ref{eq ACD_2}) with true
parameter value denoted by $\theta_{0}$, if: (i) $\{\varepsilon_{i}\}$ is an
i.i.d. sequence of r.v.s with support $(0,\infty)$, pdf $f_{\varepsilon
}\left( \cdot\right) $ bounded away from zero on compact subsets of $\left(
0,\infty\right) $, $\mathbb{E}[\varepsilon_{i}]=1$ and $\mathbb{E}
[\varepsilon_{i}^{2}]<\infty$, (ii) $\alpha_{0}+\beta_{0}<1$, then with
$\theta_{0}$ an interior point, the maximizer of $L_{n(T)}\left(
\theta\right) $ in (\ref{eq: EACD likelihood}) will be consistent and
asymptotically normal at the standard $\sqrt{T}$-rate, with a covariance
matrix given by $\left( 1/\mu\right) \Omega_{I}^{-1}\Omega_{S}\Omega
_{I}^{-1}$. Here $\mu=\mathbb{E}\left( x_{i}\right) <\infty$, and
$\Omega_{I}=\mathbb{E[}\zeta_{i}]$, $\Omega_{S}=\mathbb{E[}\xi_{i}\xi
_{i}^{\prime}]$ are given by (\ref{eq LLN for deterministic n})\ and
(\ref{eq CLT for deterministic n}) respectively.
\end{theorem}
\begin{remark}
Theorem \ref{thm main} can be extended by replacing the i.i.d. condition (i)
with the milder assumption that $\{\varepsilon_{i}\}$ is strictly stationary
and ergodic with (conditional) mean one, see Lee and Hansen (1994).
\end{remark}
\section*{References}
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\smallskip\noindent\textsc{Engle, R.F. and Russell, J.R.}\ (1998)
Autoregressive conditional duration: a new model for irregularly spaced
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