Asymptotics for the Generalized Autoregressive Conditional Duration Model

TL;DR

This paper investigates the asymptotic properties of the generalized autoregressive conditional duration (ACD) model, highlighting the importance of finite mean durations for consistency and normality of estimators, extending prior GARCH-based results.

Contribution

It establishes that strict stationarity and ergodicity are insufficient for the ACD model's estimator consistency, emphasizing the need for finite mean durations.

Findings

Finite mean durations are necessary for estimator consistency.

Previous GARCH-based assumptions are inadequate for ACD models.

Additional conditions ensure asymptotic normality of estimators.

Abstract

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.