A Comment on: “Autoregressive Conditional Duration: A New Model for Irregularly Spaced Transaction Data”
基于更新过程理论，重新审视了ACD模型QMLE的渐近性质，发现其一致性需要更强的有限期望条件，并给出了一个MLE渐近混合正态且收敛速度较慢的反例。
Based on the GARCH literature, Engle and Russell (1998) established consistency and asymptotic normality of the QMLE for the autoregressive conditional duration (ACD) model, assuming strict stationarity and ergodicity of the durations. Using novel arguments based on renewal process theory, we show that their results hold under the stronger requirement that durations have finite expectation. However, we demonstrate that this is not always the case under the assumption of stationary and ergodic durations. Specifically, we provide a counterexample where the MLE is asymptotically mixed normal and converges at a rate significantly slower than usual. The main difference between ACD and GARCH asymptotics is that the former must account for the number of durations in a given time span being random. As a by‐product, we present a new lemma which can be applied to analyze asymptotic properties of extremum estimators when the number of observations is random.