**4. The GBS-AACD Models**

Now, we introduce a generalization of the linear form for the conditional median dynamics based on the Box-Cox transformation; see Box and Cox (1964) and Fernandes and Grammig (2006) for pertinent details. Hereafter, we use the log-linear form *σi* given in (16) with *r* = 1 and *s* = 1 (i.e., the GBS-ACD(*r* = 1,*s* = 1) model, which we abbreviate as the GBS-ACD model, since a higher-order model does not increase the distributional fit of the residuals (Bhatti 2010)). Therefore, (16) results in

$$
\ln \sigma\_{i} = \alpha + \beta \ln \sigma\_{i-1} + \gamma \left[ \frac{x\_{i-1}}{\sigma\_{i-1}} \right]. \tag{23}
$$

The asymmetric version of the GBS-ACD model—GBS-AACD model—is given by

$$
\sigma\_i = \mathfrak{a} + \beta \sigma\_{i-1} + \gamma \sigma\_{i-1} \left( |\varphi\_{i-1} - b| + c(\varphi\_{i-1} - b) \right), \tag{24}
$$

where *b* and *c* are the shift and rotation parameters, respectively. By applying the Box-Cox transformation with parameter *λ* ≥ 0 to the conditional duration model process *σi* and introducing the parameter *ν*, we can write (24) as

$$\frac{\sigma\_{i}^{\lambda} - 1}{\lambda} = \kappa\_{\*} + \beta \frac{\sigma\_{i-1}^{\lambda} - 1}{\lambda} + \gamma\_{\*} \sigma\_{i-1}^{\lambda} \left( |\varphi\_{i-1} - b| + c(\varphi\_{i-1} - b) \right)^{\vee}.\tag{25}$$

The parameter *λ* determines the shape of the transformation, i.e., concave ( *λ* ≤ 1) or convex (*λ* ≥ 1), and the parameter *ν* aims to transform the (potentially shifted and rotated) term |*ϕi*−<sup>1</sup> − *b*| + *<sup>c</sup>*(*ϕi*−<sup>1</sup> − *b*) . Setting *α* = *λα*∗ − *β* + 1 and *γ* = *λγ*<sup>∗</sup>, we obtain

$$
\sigma\_i^{\lambda} = a + \beta \sigma\_{i-1}^{\lambda} + \gamma \sigma\_{i-1}^{\lambda} \left( |\varphi\_{i-1} - b| + c(\varphi\_{i-1} - b) \right)^{\nu}. \tag{26}
$$

We present below the forms of GBS-AACD models obtained from different specifications. Note that the Logarithmic GBS-ACD type II is equivalent to (23).

• Augmented ACD (GBS-AACD):

$$
\sigma\_i^{\lambda} = a + \beta \sigma\_{i-1}^{\lambda} + \gamma \sigma\_{i-1}^{\lambda} \left( |\varphi\_{i-1} - b| + c (\varphi\_{i-1} - b) \right)^{\nu}.
$$

• Asymmetric power ACD (GBS-A-PACD) ( *λ* = *ν*):

$$
\sigma\_i^{\lambda} = a + \beta \sigma\_{i-1}^{\lambda} + \gamma \sigma\_{i-1}^{\lambda} \left( |\varrho\_{i-1} - b| + \mathfrak{c} (\varrho\_{i-1} - b) \right)^{\lambda}.
$$

• Asymmetric logarithmic ACD (GBS-A-LACD) ( *λ* → 0 and *ν* = 1):

$$
\ln \sigma\_{\bar{i}} = a + \beta \ln \sigma\_{\bar{i}-1} + \gamma \left( |\varphi\_{\bar{i}-1} - b| + c(\varphi\_{\bar{i}-1} - b) \right).
$$

• Asymmetric ACD (GBS-A-ACD) ( *λ* = *ν* = 1):

$$
\sigma\_i = \alpha + \beta \sigma\_{i-1} + \gamma \sigma\_{i-1} \left( |\varphi\_{i-1} - b| + c(\varphi\_{i-1} - b) \right).
$$

• Power ACD (GBS-PACD) ( *λ* = *ν* and *b* = *c* = 0):

$$
\sigma\_i^{\lambda} = \mathfrak{a} + \beta \sigma\_{i-1}^{\lambda} + \gamma \mathbf{x}\_{i-1}^{\lambda}.
$$

• Box-Cox ACD (GBS-BCACD) ( *λ* → 0 and *b* = *c* = 0):

$$
\ln \sigma\_{\bar{i}} = \alpha + \beta \ln \sigma\_{\bar{i}-1} + \gamma q\_{\bar{i}-1}^{\vee}.
$$

• Logarithmic ACD type I (GBS-LACD I) ( *λ*, *ν* → 0 and *b* = *c* = 0):

$$
\ln \sigma\_{\bar{i}} = \alpha + \beta \ln \sigma\_{\bar{i}-1} + \gamma \ln x\_{\bar{i}-1}.
$$

• Logarithmic ACD type II (GBS-LACD II) ( *λ* → 0, *ν* = 1 and *b* = *c* = 0):

$$
\ln \sigma\_{\bar{i}} = \alpha + \beta \ln \sigma\_{\bar{i}-1} + \gamma q\_{\bar{i}-1} \cdot \sigma\_{\bar{i}}
$$

#### **5. Numerical Results for the GBS-ACD Models**

In this section, we perform two simulation studies, one for evaluating the behavior of the ML estimators of the GBS-ACD models, and another for examining the performance of the residuals. We have focused on the GBS-ACD models because similar results were obtained for the GBS-AACD models.
