*5.2. Study of Residuals*

We now carry out an MC simulation study to examine the performance of the Cox–Snell residual *r*cs defined in (22). To do so, we use the estimation procedure presented in Section 5.1 and consider only the BS-PE-ACD model, as it provides greater flexibility in relation to other models, that is, it has either less or greater (lighter or heavier tails) than the BS distribution. The BS-PE-ACD samples are generated using the transformation in (5). We simulate *B* = 1000 MC samples of size *n* = 500. The empirical autocorrelation function (ACF) of the residual *r*cs is plotted in Figure 1a. This plot indicates that the BS-PE-ACD model is well specified, since the residual *r*cs mimics a sequence of independent and identically distributed RVs and there is no indication of serial correlation. Moreover, the empirical mean of the residual *r*cs, whose value was expected to be 1, was 0.9836. Finally, using a quantile-against-quantile (QQ) plot with a simulated envelope (see Figure 1b), we note that the

Cox–Snell residual has an excellent agreemen<sup>t</sup> with the EXP(1) distribution, which supports the adequacy and flexibility of the BS-PE-ACD model. It is then possible to conclude that the residual *r*cs seems adequate to assess the adjustment of the proposed models.

**Figure 1.** Autocorrelation function (ACF) plot and quantile-against-quantile (QQ) plot with envelope for the residuals.

#### **6. Application to Analysis of Financial Transaction Data**

In this section, our objective is to assess the GBS-ACD and GBS-AACD models using TD data. In particular, we consider here three TD data sets studied in Bhatti (2010), corresponding to the time elapsed (in seconds) between two consecutive transactions, which cover forty trading days from January 1, 2002 to February 28, 2002: International Business Machines (IBM), Johnson and Johnson Company (JNJ), and The Proctor and Gamble Company (PG). Note that, as mentioned before, these types of data exhibit some diurnal patterns, so that the final data sets are constructed from adjusted TD *x*¯*i* = *xi*/*φ*ˆ, where *φ*ˆ = exp(*s*<sup>ˆ</sup>) and *s*ˆ denotes a set of quadratic functions and indicator variables for each half-hour interval of the trading day from 9:30 am to 4:00 pm; for more details, see Giot (2000), Tsay (2002), and Bhatti (2010).
