*6.1. Exploratory Data Analysis*

Table 3 provides some descriptive statistics for both plain and diurnally adjusted TD data, which include central tendency statistics and coefficients of variation (CV), of skewness (CS), and of kurtosis (CK), among others. These measures indicate the positively skewed nature and the high kurtosis of the data. Figure 2 shows graphical plots of the ACF and partial ACF for the IBM, JNJ, and PG data sets, which indicate the presence of serial correlation.


**Table 3.** Summary statistics for the International Business Machines (IBM), Johnson and Johnson Company (JNJ), and The Proctor and Gamble Company (PG) data sets.

**Figure 2.** Autocorrelation and partial autocorrelation functions for the indicated data sets.

The hazard function of a positive RV *X* is given by *hX*(*t*) = *fX*(*x*)/(<sup>1</sup> − *FX*(*x*)), where *fX*(·) and *FX*(·) are the PDF and CDF of *X*, respectively. A useful way to characterize the hazard function is by the scaled total time on test (TTT) function, namely, we can detect the type of hazard function that the data have and then choose an appropriate distribution. The TTT function is given by *WX*(*u*) = *H*−<sup>1</sup> *X* (*u*)/*H*−<sup>1</sup> *X* (1) for 0 ≤ *u* ≤ 1, where *H*−<sup>1</sup> *X* (*u*) = *F*−<sup>1</sup> *X* (*u*) 0 [1 − *FX*(*y*)]d*y*, where *F*−<sup>1</sup> *X* (·) is the inverse CDF of *X*. By plotting the consecutive points (*k*/*<sup>n</sup>*, *Wn*(*k*/*n*)) with *Wn*(*k*/*n*)=[∑*ki*=<sup>1</sup> *<sup>x</sup>*(*i*) + (*n* − *k*)*xk*]/ ∑*ni*=<sup>1</sup> *<sup>x</sup>*(*i*) for *k* = 0, ... , *n*, and *<sup>x</sup>*(*i*) being the *i*th-order statistic, it is possible to approximate *WX*(·); see Aarset (1987) and Azevedo et al. (2012).

From Figure 3, we observe that the TTT plots sugges<sup>t</sup> a failure rate with a unimodal shape. We also observe that the histograms sugges<sup>t</sup> a positive skewness for the data density. This supports the results obtained in Table 3. However, Huber and Vanderviere (2008) pointed out that, in cases where the data follow a skewed distribution, a significant number of observations can be classified as atypical when they are not. The boxplots depicted in Figure 3 sugges<sup>t</sup> such a situation, i.e., most of the observations considered as potential outliers by the usual boxplot are not outliers when we consider the adjusted boxplot.

**Figure 3.** Total time on test (TTT) plot (**a**), histogram (**b**), and usual and adjusted boxplots (**c**) for the indicated data sets.

#### *6.2. Estimation Results and Analysis of Goodness-of-Fit for the GBS-ACD Models*

We now estimate the GBS-ACD models by the maximum likelihood method using the steps described in Section 5.1. Tables 4–6 present the estimation results for the indicated models. The standard errors (SEs) are reported in parentheses and stands for the value of the log-likelihood function, whereas AIC = −2 +2*k* and BIC = −2 + *k* ln *n* denote, respectively, the Akaike information and Bayesian information criteria, where *k* stands for the number of parameters and *n* for the number of observations. The maximum and minimum values of the sample autocorrelations (ACF) from order 1 to 60 are also reported. Finally, *γ*¯ denotes the mean magnitude of autocorrelation for the first 15 lags, namely, *γ*¯ = 1/15 ∑<sup>15</sup>*i*=<sup>1</sup> |*<sup>γ</sup>k*|, where *γk* = cor(*xi*, *xi*+*<sup>k</sup>*). The mean magnitude of autocorrelation *γ*¯ is relevant for separating the influence of the sample size on the measure of the degree of autocorrelation in the residuals.

From Tables 4–6, we observe that all of the parameters are statistically significant at the 1% level. It is also interesting to observe that, in general, the ACD parameter estimates are very similar across

the models independently of the assumed distribution. In terms of AIC values, the BS-PE-ACD model outperforms all other models. Based on the BIC values, we note that the BS-PE-ACD model once again outperforms the remaining models, except for the JNJ data set. However, in this case, there does not exist one best model, since the BIC values for the BS-ACD and BS-PE-ACD models are very close.

In order to check for misspecification, we look at the sample ACF from order 1 to 60. Tables 4–6 report that there is no sample autocorrelation greater than 0.05 (in magnitude) throughout the models and residuals. Figure 4 shows the QQ plots of the Cox–Snell residual with the IBM, JNJ, and PG data sets. The QQ plot allows us to check graphically if the residual follows the EXP(1) distribution. These graphical plots show an overall superiority in terms of fit of the BS-PE-ACD model. Moreover, the empirical means of the residual *r*cs for the BS-PE-ACD model with the IBM, GM, and PG data sets were 1.0271, 0.9990, and 1.0153, respectively. Thus, the BS-PE-ACD model seems to be more suitable for modeling the data considered. It must be emphasized that this model provides greater flexibility in terms of kurtosis compared to the BS-ACD model.

**Table 4.** Estimation results based on the generalized Birnbaum–Saunders autoregressive conditional duration (GBS-ACD) models for IBM trade durations.


**Table 5.** Estimation results based on the GBS-ACD models for JNJ trade durations.



**Table 6.** Estimation results based on the GBS-ACD models for PG trade durations.

**Figure 4.** QQ plot for the Cox–Snell residual with the indicated data sets.

#### *6.3. Estimation Results for the BS-PE-AACD Models*

We estimate here different ACD specifications (see Section 4) assuming a BS-PE PDF and using JNJ TD data. We focus on the BS-PE-AACD models (in short, AACD models), since, as observed in Section 6.2, this model fits the data adequately to provide effective ML-based inference. The estimation is performed using the steps presented in Section 5.1.

Tables 7 and 8 report the estimation results for different specifications. It is important to point out that the estimates of the BS-PE parameters *κ* and *η* are quite robust throughout the specifications. The Box-Cox ACD result (see column BCACD) shows that allowing *ν* of *ϕi*−<sup>1</sup> to freely vary in the logarithm ACD processes (LACD I and LACD II) increases the log-likelihood value, indicating that *ν* may play a role. In fact, *ν*ˆ is significantly different from zero and one, thus supporting the BCACD model against its logarithm counterparts, i.e., LACD I and LACD II. The AIC values show that the BCACD, LACD I, and AACD are best models. From the BIC values, the LACD I, BCACD, and AACD models are the best ones. Note, however, that the BIC values for the LACD I and BCACD models are quite close. Tables 4–6 also show that there is no sample autocorrelation greater than 0.05 (in magnitude) throughout the models and residuals.


**Table 7.** Estimation results for ACD specifications of JNJ trade durations. A star (∗) indicates that the parameter estimate is not significantly different from zero.

**Table 8.** Estimation results for ACD specifications of JNJ trade durations. A star (∗) indicates that the parameter estimate is not significantly different from zero.

