Asymptotics for the Generalized Autoregressive Conditional Duration Model

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Engle and Russell (1998, 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.

\noindentKeywords:\ autoregressive conditional duration (ACD); quasi maximum likelihood.

Introduction

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

align[align omitted — 233 chars of source]

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,

equation[equation omitted — 237 chars of source]

with $\omega,\alpha$ and $\beta$ positive, and initial values $(x_{0},\psi _{0}(\theta))^{\prime}=\gamma$.

ER\ note that the likelihood function in ((ref)) 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 $\theta =\theta_{0}$:

align[align omitted — 591 chars of source]

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 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

align[align omitted — 283 chars of source]

does not imply that their random $n(T)$ analogues in ((ref) )-((ref)) 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 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

equation[equation omitted — 150 chars of source]

This in turn (as $n(T)\rightarrow\infty$ a.s.) is sufficient for the deterministic $n$ LLN in ((ref)) to imply that its random $n$ analogue ((ref)) holds. To establish the random $n$ CLT in ((ref)), we replace the deterministic $n$ CLT\ in ((ref)) 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.

Main result

In order to derive the asymptotic distribution of the QMLE\ for the ACD model given by ((ref))-((ref)) we first introduce the following general lemma, which extends the results in Lee and Hansen (1994) to allow for a random number of observations.

lemmaConsider $Q_{n}(\varphi)\in\mathbb{R}$, 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 $n\rightarrow\infty$:$\smallskip$ $ \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) & $-n^{-1}\partial^{2}Q_{n}(\varphi_{0})/\partial\varphi\partial \varphi^{\prime}\overset{a.s.}{\rightarrow}\Omega_{I}>0$,\\ (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$ 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:$\smallskip$ $ \begin{tabular} [c]{lc} (C.4) & As $T\rightarrow\infty$, $n\left( T\right) /T\overset{a.s. }{\rightarrow}c$. \end{tabular} \ \smallskip$ 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:$\smallskip$ $ \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} \ $

The proof of Lemma (ref) 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.

theoremFor the ACD\ model ((ref))-((ref)) 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)) 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))\ and ((ref)) respectively.

remarkTheorem (ref) 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).

References

\noindentCavaliere, G., Mikosch, T., Rahbek A., and Vilandt, F. (2022) The econometrics of financial duration modeling. arXiv:2208.02098.

\noindentEngle, R.F. and Russell, J.R.\ (1998) Autoregressive conditional duration: a new model for irregularly spaced transaction data. Econometrica, 66:1127--1162.

\noindentGut, A.\ (2009) Stopped Random Walks: Limit Theorems and Applications. Springer, NY.

\noindentJensen, S.T. and Rahbek, A. (2004) Asymptotic inference for nonstationary GARCH. Econometric Theory, 20:1203--1226.

\noindentJensen, S.T. and Rahbek, A. (2007) On the law of large numbers for (geometrically) ergodic Markov chains. Econometric Theory, 23:761--766.

\noindentLee, S. and Hansen, B.\ (1994) Asymptotic theory for the GARCH$(1,1)$\ quasi-maximum likelihood estimator. Econometric Theory, 10:29--52.

\noindentMeitz, M. and Saikkonen, P.\ (2008) Ergodicity, mixing, and existence of moments of a class of Markov models with applications to GARCH and ACD models. Econometric Theory, 24:1291--1320.

\noindentPhillips, P.C.B. and Solo, V. (1992) Asymptotics for linear processes. The Annals of Statistics, 20:971--1001.