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.

翻译：Engle和Russell(1998, Econometrica, 66:1127–1162)将GARCH文献中的结论应用于广义自回归条件持续时间(ACD)模型(即所谓的ACD(1,1)模型),在严格平稳性和遍历性假设下证明了(指数)准极大似然估计量(QMLE)的一致性和渐近正态性。然而,GARCH方法未考虑给定观测时段内持续时间的数量具有随机性。因此,与Engle和Russell(1998)不同,我们证明仅依靠严格平稳性和遍历性不足以保证一致性和渐近正态性,并提出了额外充分条件以应对持续时间的随机数量。具体而言,我们论证了持续时间需满足更强的要求,即其均值有限。