*3.1. Existing ACD Models*

Let *Xi* = *Ti* − *Ti*−<sup>1</sup> denote the trade duration, i.e., the time elapsed between two transactions occurring at times *Ti* and *Ti*−1. Engle and Russell (1998) assumed that the serial dependence commonly found in financial duration data is described by *ψi* = <sup>E</sup>[*Xi*|F*<sup>i</sup>*−<sup>1</sup>], where *ψi* stands for the conditional mean of the *i*th trade duration based on the conditioning information set F*i*−1, which includes all information available at time *Ti*−1. The basic ACD(*<sup>r</sup>*,*<sup>s</sup>*) model is then defined as

$$\begin{array}{rcl} X\_i &=& \psi\_i \,\varepsilon\_i \\ \psi\_i &=& \alpha + \sum\_{j=1}^r \beta\_j \psi\_{i-j} + \sum\_{j=1}^s \gamma\_j \mathbf{x}\_{i-j}, \quad i = 1, \dots, n, \end{array} \tag{12}$$

where *r* and *s* refer to the orders of the lags and {*<sup>ε</sup>i*} is an independent and identically distributed nonnegative sequence with PDF *f*(·). Engle and Russell (1998) assumed a linear ACD(1,1) model defined by *ψi* = *α* + *βxi*−<sup>1</sup> + *γψ<sup>i</sup>*−1, where *α*, *β*, and *γ* are the parameters. Note that a wide range of ACD model specifications may be defined by different distributions of *εi* and specifications of *ψi*; see Fernandes and Grammig (2006) and Pathmanathan et al. (2009).

An alternative ACD model is the Birnbaum–Saunders autoregressive conditional duration (BS-ACD) model proposed by Bhatti (2010). This approach takes into account the natural scale parameter in the BS(*θ*) distribution to specify the BS-ACD model in terms of a time-varying conditional median duration *σi* = *F*−<sup>1</sup> BS (0.5|F*<sup>i</sup>*−<sup>1</sup>), where *F*BS denotes the CDF of the BS distribution. This specification has several advantages over the existing ACD models, as previously mentioned, including a realistic distributional assumption—an expected improvement in the model fit as a result of the natural parametrization in terms of the conditional median duration, since the median is generally considered to be a better measure of central tendency than the mean for asymmetrical and heavy-tailed distributions—and ease of fitting.

The PDF associated with the BS-ACD(*<sup>r</sup>*,*<sup>s</sup>*) model is given by

> ln

$$f\_{\rm BS}(\mathbf{x}\_{i};\theta\_{i}) = \phi\left(a(\mathbf{x}\_{i};\theta\_{i})\right)a'(\mathbf{x}\_{i};\theta\_{i}), \quad i = 1,\ldots,n,\tag{13}$$

where

$$
\theta\_i = (\kappa, \sigma\_i)^\top, \quad i = 1, \dots, n,
$$

$$
\sigma\_i = \alpha + \sum\_{j=1}^r \beta\_j \ln \sigma\_{i-j} + \sum\_{j=1}^s \gamma\_j \lceil \frac{x\_{i-j}}{\sigma\_{i-j}} \rceil. \tag{14}
$$
