Autoregressive conditional duration

In financial econometrics, an autoregressive conditional duration (ACD, Engle and Russell (1998)) model considers irregularly spaced and autocorrelated intertrade durations. ACD is analogous to GARCH. In a continuous double auction (a common trading mechanism in many financial markets) waiting times between two consecutive trades vary at random.

Let ${\displaystyle ~\tau _{t}~}$ denote the duration (the waiting time between consecutive trades) and assume that ${\displaystyle ~\tau _{t}=\theta _{t}z_{t}~}$, where ${\displaystyle z_{t}}$ are independent and identically distributed random variables, positive and with ${\displaystyle \operatorname {E} (z_{t})=1}$ and where the series ${\displaystyle ~\theta _{t}~}$ is given by:

${\displaystyle \theta _{t}=\alpha _{0}+\alpha _{1}\tau _{t-1}+\cdots +\alpha _{q}\tau _{t-q}+\beta _{1}\theta _{t-1}+\cdots +\beta _{p}\theta _{t-p}=\alpha _{0}+\sum _{i=1}^{q}\alpha _{i}\tau _{t-i}+\sum _{i=1}^{p}\beta _{i}\theta _{t-i}}$

and where ${\displaystyle ~\alpha _{0}>0~}$, ${\displaystyle \alpha _{i}\geq 0}$, ${\displaystyle \beta _{i}\geq 0}$, ${\displaystyle ~i>0}$.