**1. Introduction**

The modeling of high-frequency financial data has been the focus of intense interest over the last decades. A prominent approach to modeling the durations between successive events (trades, quotes, price changes, etc.) was introduced by Engle and Russell (1998). These authors proposed the autoregressive conditional duration (ACD) model, which has some similarities with the ARCH (Engle 1982) and GARCH (Bollerslev 1986) models. The usefulness of appropriately modeling duration data is stressed by the relatively recent market microstructure literature; see Diamond and Verrechia (1987), Easley and O'Hara (1992), and Easley et al. (1997). Generalizations of the original ACD model are basically based on the following three aspects, i.e., (a) the distributional assumption in order to yield a unimodal failure rate (FR) (Grammig and Maurer 2000; Lunde 1999), (b) the linear form for the conditional mean (median) dynamics (Allen et al. 2008; Bauwens and Giot 2000; Fernandes and Grammig 2006), and (c) the time series properties (Bauwens and Giot 2003; Chiang 2007; De Luca and Zuccolotto 2006; Jasiak 1998; Zhang et al. 2001); see the reviews by Pacurar (2008) and Bhogal and Variyam Thekke (2019). Bhatti (2010) proposed a generalization of the ACD model that falls into all three branches above, based on the Birnbaum–Saunders (BS) distribution, denoted as the BS-ACD model. This model has several advantages over the traditional ACD ones; in particular, the BS-ACD model (1) has a realistic distributional assumption, that is, it provides both an asymmetric probability

density function (PDF) and a unimodal FR shape; (2) it provides a natural parametrization of the point process in terms of a conditional median duration which is expected to improve the model fit despite a conditional mean 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 (3) has easy implementation for estimation; see Ghosh and Mukherjee (2006), Bhatti (2010), Leiva et al. (2014), and Saulo et al. (2019).

Based on the relationship between the BS and symmetric distributions, Díaz-García and Leiva (2005) introduced generalized BS (GBS) distributions, obtaining a wider class of distributions that has either lighter or heavier tails than the BS density, allowing them to provide more flexibility. This new class essentially provides flexibility in the kurtosis level; see Sanhueza et al. (2008). In addition, the GBS distributions produce models whose parameter estimates are often robust to atypical data; see Leiva et al. (2008) and Barros et al. (2008). The GBS family includes as special cases the BS, BS-Laplace (BS-LA), BS-Logistic (BS-LO), BS-power-exponential (BS-PE), and BS-Student-*t* (BS-*t*) distributions.

The main aim of this work is to generalize the BS-ACD model, which was proposed by Bhatti (2010), based on GBS distributions (GBS-ACD). The proposed models should hold with the properties of the BS-ACD model, but, in addition, they should provide further properties and more flexibility. As mentioned before, the GBS family has models that have heavier tails than the BS density, and this characteristic is very useful in the modeling of high-frequency financial durations, since duration data are heavy-tailed and heavily right-skewed. We subsequently develop a class of augmented GBS-ACD (GBS-AACD) models by using the Box-Cox transformation (Box and Cox 1964) with a shape parameter *λ* ≥ 0 to the conditional duration process and an asymmetric response to shocks; see Fernandes and Grammig (2006). Thus, the proposed GBS-ACD and GBS-AACD models would provide greater range and flexibility while fitting data. We apply the proposed models to high-frequency financial transaction (trade duration, TD) data. This type of data has unique features absent in data with low frequencies. For example, TD data (1) inherently arrive in irregular time intervals, (2) possess a large number of observations, (3) exhibit some diurnal pattern, i.e., activity is higher near the beginning and closing than in the middle of the trading day, and (4) present a unimodal failure rate; see Engle and Russell (1998) and Bhatti (2010). In addition, TD data have a relevant role in market microstructure theory, since they can be used as proxies for the existence of information in the market, and then serve as predictors for other market microstructure variables; see Mayorov (2011).

The rest of the paper proceeds as follows. Section 2 describes the BS and GBS distributions. In addition, some propositions are presented. Section 3 derives the GBS-ACD models associated with these distributions. Section 4 derives the GBS-AACD class of models. A Monte Carlo study of the proposed GBS-ACD model is performed in Section 5. Next, Section 6 presents an application of the proposed models to three data sets of New York stock exchange (NYSE) securities, and their fits are then assessed by a goodness-of-fit test. Finally, Section 7 offers some concluding remarks.

#### **2. The Birnbaum–Saunders Distribution and Its Generalization**

In this section, we describe the BS and GBS distributions and some of their properties.

The two-parameter BS distribution was introduced by Birnbaum and Saunders (1969) for modeling failure times of a material exposed to fatigue. They assumed that the fatigue failure follows from the development and growth of a dominant crack. Let *θ* = (*<sup>κ</sup>*, *σ*) and

$$a(\mathbf{x}; \boldsymbol{\theta}) = \frac{1}{\kappa} \left[ \sqrt{\frac{\mathbf{x}}{\sigma}} - \sqrt{\frac{\sigma}{\mathbf{x}}} \right], \quad \mathbf{x} > 0 \text{ and } \kappa, \sigma > 0. \tag{1}$$

Expressions for the first, second, and third derivatives of the function *<sup>a</sup>*(·; *θ*) are, respectively, given by

$$a'(\mathbf{x}; \boldsymbol{\theta}) = \frac{1}{2\kappa r} \left[ \left(\frac{\sigma}{x}\right)^{1/2} + \left(\frac{\sigma}{x}\right)^{3/2} \right], \quad a''(\mathbf{x}; \boldsymbol{\theta}) = -\frac{1}{4\kappa rx} \left[ \left(\frac{\sigma}{x}\right)^{1/2} + 3\left(\frac{\sigma}{x}\right)^{3/2} \right],\tag{2}$$

$$a'''(\mathbf{x}; \boldsymbol{\theta}) = \frac{3}{8\kappa rx^2} \left[ \left(\frac{\sigma}{x}\right)^{1/2} + 5\left(\frac{\sigma}{x}\right)^{3/2} \right].$$

A random variable (RV) *X* has a BS distribution with parameter vector *θ* = (*<sup>κ</sup>*, *<sup>σ</sup>*), denoted by BS(*θ*), if it can be expressed as

$$X = a^{-1}(Z; \theta) = \frac{\sigma}{4} \left[ \kappa Z + \sqrt{(\kappa Z)^2 + 4} \right]^2, \quad Z \sim \text{N}(0, 1), \tag{3}$$

where *<sup>a</sup><sup>−</sup>***<sup>1</sup>**(·; *θ*) denotes the inverse function of *<sup>a</sup>*(·; *<sup>θ</sup>*), *κ* is a shape parameter, and when it decreases to zero, the BS distribution approaches the normal distribution with mean *σ* and variance *τ*, such that *τ* → 0 when *κ* → 0. In addition, *σ* is a scale parameter and also the median of the distribution *<sup>F</sup>*BS(*σ*) = 0.5, where *F*BS is the BS cumulative distribution function (CDF). The BS distribution holds proportionality and reciprocal properties given by *b X* ∼ BS(*<sup>κ</sup>*, *b <sup>σ</sup>*), with *b* > 0, and 1/*X* ∼ BS(*<sup>κ</sup>*, 1/*σ*); see Saunders (1974). The probability density function (PDF) of a two-parameter BS random variable *X* is given by

$$f\_{\rm BS}(\mathbf{x};\boldsymbol{\theta}) = \phi(a(\mathbf{x};\boldsymbol{\theta}))a'(\mathbf{x};\boldsymbol{\theta}), \quad \mathbf{x} > \mathbf{0}, \tag{4}$$

where *φ*(·) denotes the PDF of the standard normal distribution.

Díaz-García and Leiva (2005) proposed the GBS distribution by assuming that *Z* in (3) follows a symmetric distribution in R, denoted by *X* ∼ GBS(*<sup>θ</sup>*, *g*), where *g* is a density generator function associated with the particular symmetric distribution. An RV *Z* has a standard symmetric distribution, denoted by *Z* ∼ S(0, 1; *g*) ≡ <sup>S</sup>(*g*), if its density takes the form *fZ*(*z*) = *c g*(*z*<sup>2</sup>) for *z* ∈ R, where *g*(*u*) with *u* > 0 is a real function that generates the density of *Z*, and *c* is the normalization constant, that is, *c* = 1/ +∞ −∞ *<sup>g</sup>*(*z*<sup>2</sup>)*dz*. Note that *U* = *Z*<sup>2</sup> ∼ <sup>G</sup>*χ*<sup>2</sup>(1; *g*), namely, *U* has a generalized chi-squared (G*χ*2) distribution with one degree of freedom and density generator *g*; see Fang et al. (1990). Table 1 presents some characteristics and the values of *<sup>u</sup>*1(*g*), *<sup>u</sup>*2(*g*), *<sup>u</sup>*3(*g*), and *<sup>u</sup>*4(*g*) for the Laplace, logistic, normal, power-exponential (PE) and Student-*t* symmetric distributions, where *ur*(*g*) = E[*Ur*] denotes the *r*th moment of *U*.

**Table 1.** Constants (*c* and *cg*2 ), density generators (*g*), and expressions of some moments *ur*(*g*) for the indicated distributions.


Consider an RV *Z* such that *Z* = *a*(*<sup>X</sup>*; *θ*) ∼ <sup>S</sup>(*g*) so that

$$X = a^{-1}(Z; \theta) \sim \text{GBS}(\theta, \lg). \tag{5}$$

The density associated with *X* in (5) is given by

$$f\_{\rm GBS}(\mathbf{x}; \boldsymbol{\theta}, \mathbf{g}) = c \, \mathrm{g}\left(a^2(\mathbf{x}; \boldsymbol{\theta})\right) a'(\mathbf{x}; \boldsymbol{\theta}), \quad \mathbf{x} > \mathbf{0}, \tag{6}$$

where, as mentioned earlier, *g* is the generator and *c* the normalizing constant associated with a particular symmetric density; see Table 1. The mean and variance of *X* are, respectively,

$$\mathrm{E}[X] = \frac{\sigma}{2}(2 + \mu\_1 \kappa^2), \quad \mathrm{Var}[X] = \frac{\sigma^2 \kappa^2}{4}(2\kappa^2 \mu\_2 + 4\mu\_1 - \kappa^2 \mu\_1^2), \tag{7}$$

where *ur* = *ur*(*g*) = <sup>E</sup>[*Ur*], with *U* ∼ <sup>G</sup>*χ*<sup>2</sup>(1, *g*); see Table 1.

Based on Table 1, the expressions for the BS-LA, BS-LO, BS-PE, and BS-*t* densities are as follows:

*f*BS−LA(*<sup>x</sup>*; *θ*) = 14*κσ* exp −1*κ* - *xσ* − *σx σx* 1/2 + *σx* 3/2 , *f*BS−L0(*<sup>x</sup>*; *θ*) = 12*κσ* exp( 1*κ* [√ *xσ* −√*<sup>σ</sup>x* ]) [<sup>1</sup>+exp( 1*κ* [√ *xσ* −√*<sup>σ</sup>x* ])]<sup>2</sup> *σx* 1/2 + *σx* 3/2 , *f*BS−PE(*<sup>x</sup>*; *θ*, *η*) = *η* Γ 12*η* 2 12*η* +1*κσ* exp − 1 2*κ*2*<sup>η</sup>* & *xσ* + *σx* − 2'*<sup>η</sup> σx* 1/2 + *σx* 3/2 , *f*BS−*<sup>t</sup>*(*<sup>x</sup>*; *θ*, *η*) = Γ *η*+1 2 <sup>2</sup>√*ηπ* Γ( *η*2 )*κσ* 1 + 1*ηκ*<sup>2</sup> *xσ* + *σx* − 2<sup>−</sup> *η*+1 2 *σx* 1/2 + *σx* 3/2 , *x* > 0 and *κ*, *σ*, *η* > 0.

Note that if *η* = 1 (BS-PE) or if *η* → ∞ (BS-*t*), then we obtain the BS distribution. It is worthwhile to point out that the BS-PE distribution has a greater (smaller) kurtosis than that of the BS distribution when *η* < 1 (*η* > 1). In addition, the BS-*t* distribution has a greater degree of kurtosis than that of the BS distribution for *η* > 8; see Marchant et al. (2013).

Let

$$b(h; \kappa, \iota, \phi) = \frac{2}{\kappa} \sinh \left( \frac{h - \iota}{\phi} \right), \quad h \in \mathbb{R}, \ \kappa, \phi > 0 \text{ and } \iota \in \mathbb{R}. \tag{8}$$

An alternative way to obtain GBS distributions is through sinh-symmetric (SHS) distributions. Díaz-García and Domínguez-Molina (2006) proposed SHS distributions by using the sinh-normal distribution introduced by Rieck and Nedelman (1991) in the symmetric case. They assumed the standard symmetrically distributed RV *Z* as follows:

$$Z = b(H; \mathfrak{x}, \mathfrak{t}, \mathfrak{\phi}) \sim \mathcal{S}(\mathfrak{g}).\tag{9}$$

Then,

$$H = b^{-1}(Z; \kappa, \iota, \phi) = \iota + \phi \operatorname{arcsinh}\left(\frac{\kappa Z}{2}\right) \sim \operatorname{SFIS}(\kappa, \iota, \phi, \lg). \tag{10}$$

The density associated with *H* in (10) is given by

$$f\_{\rm SHS}(h; \mathbf{x}, \iota, \boldsymbol{\phi}, \boldsymbol{\phi}) = c \, \mathrm{g} \left( b^2(h; \mathbf{x}, \iota, \boldsymbol{\phi}) \right) b'(h; \mathbf{x}, \iota, \boldsymbol{\phi}), \quad h \in \mathbb{R}, \; \mathbf{x}, \boldsymbol{\phi} > 0 \text{ and } \iota \in \mathbb{R}, \tag{11}$$

where *g* and *c* are as given in (6). A prominent result, which will be useful later on, is the following.

**Proposition 1** (See Rieck and Nedelman (1991))**.** *If H* ∼ *SHS*(*<sup>κ</sup>*, *ι* = ln *σ*, *φ* = 2, *g*)*, then X* = exp(*H*) ∼ *GBS*(*<sup>θ</sup>*, *g*)*, which is denoted by H* ∼ *log-GBS*(*<sup>κ</sup>*, *ι*, *g*)*.*
