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.

Definition
Let $$ ~\tau_t~ $$ denote the duration (the waiting time between consecutive trades) and assume that $$ ~\tau_t=\theta_t z_t ~$$, where $$ z_t $$ are independent and identically distributed random variables, positive and with $$ \operatorname{E}(z_t) = 1$$ and where the series $$ ~\theta_t~ $$ is given by:

$$ \theta_t = \alpha_0 + \alpha_1 \tau_{t-1} + \cdots + \alpha_q \tau_{t-q} + \beta_1 \theta_{t-1} + \cdots + \beta_p\theta_{t-p} = \alpha_0 + \sum_{i=1}^q \alpha_i \tau_{t-i} + \sum_{i=1}^p \beta_i \theta_{t-i} $$

and where $$ ~\alpha_0>0~ $$, $$ \alpha_i\ge 0$$, $$ \beta_i \ge 0 $$, $$~i>0$$.