We validate our retrospective monitoring procedure via three real examples in this section. For each example, we fit a Log ACD(10, 0) model using PEF approach with λ = 2.
4.5.1 BAC for A Week
As we discussed in Section 3.4.1, estimates from non-penalized estimating functions for BAC during the week of June 10, 2013 vary a lot. For Mon. June 10, Wed. June 12 and Thur. June 13, Log ACD(1, 0) were selected as the best model. Although Log ACD(1, 0) was the best fit for Tue. June 11, the estimates are different from the other three days. For Fri. June 14, a
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Log ACD(2, 2) was chosen by the non-penalized estimating functions approach. Due to this discrepancy, we want to investigate the existence of structural breaks. Assuming durations for the whole week is given, Figure 26 displays the combined durations for BAC during the week of June 10, 2013. Black dashed lines represent the boundaries between different days. Red dashed
Figure 26: Combined durations for BAC during the week of June 10, 2013. Black dashed lines represent the boundaries between different days. Red dashed lines represent the estimated locations of structural breaks
lines represent the estimated locations of structural breaks using PEF approach. Noted that the first three boundaries are correctly identified while the fourth estimated boundary departs from the real value. Also, two more estimated structural breaks (the second and fifth red dashed lines) are found. The two structural breaks are within Tue. June 11 and Thur. June 13 respectively. Instead of being two false alarms, news or analytical reports were announced on both days. On Tue. June 11 2013, PRNewswire (2013c) reported the news below
Company Director, R. David Yost, purchased his share at price of US $11.51, which was almost 15.55% return by the end of last trading session. Investors may want to find out how Bank of America insiders like CEOs, CFOs and Directors
are thinking about the future of the company.
And on Thur. June 13 2013, PRNewswire (2013a) reported the news below
By the end of last trading session, the share rose about 12.5% this year. As the US banking industry experienced a strong first quarter with record profits, the market will consider it as a recovery signal after 2008 financial crisis.
Given the above news reports, the two extra estimated structural breaks may indicate the underlying change of the market. A further investigation may be needed.
4.5.2 BAC for Three Days
Figure 27: Combined durations for BAC during the first three days of week June 17, 2013. Black dashed lines represent the boundaries between different days. Red dashed lines represent the estimated locations of structural breaks
In this example, we examine the changes occurred for BAC in three consecutive days (Mon.
Jun 17, Tue. Jun 18 and Wed. Jun 19). Abnormal Log ACD(1, 0) were selected for the first and third days while Log ACD(3, 3) were the best fit for the second day. Figure 27 displays
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the combined durations for three days. Boundaries for different days and estimated locations are marked as black dashed lines and red dashed lines respectively. The two actual boundaries are correctly identified with a reasonable difference. There are three extra estimated locations of structural breaks. The second and third dashed lines occur in Tue. June 18. Since Log ACD(3, 3) was the best model for that day, two structural breaks explains the underlying change of the stock behavior. We found that on Tue. June 18, PRNewswire (2013b) reported the following news.
Company Director, Charles K. Gifford, sold his shares at price of US$13.07 for about US$7.59 million on June 14. Investors may want to find out how Bank of America insiders like CEOs, CFOs and Directors are thinking about the future of the company.
Since stock market changes rapidly, any kinds of insider information can greatly influence the pattern of stock behavior. We didn’t find any news regarding Bank of America published on Wed. June 19. But the fifth estimated location of structural breaks divides Wed. June 19 into two halves. Long durations (frequent transactions) occurred in the morning while relatively short durations (less frequent transactions) occurred in the afternoon.
4.5.3 BAC for Two Days
The third example examines the performance of retrospective monitoring procedure when there is only one change in models between two consecutive days. In Table 2, we observe approximately the same estimates for Wed. June 5 and Thur. June 6. Similarly, we use black dashed lines to represent the boundaries between days and red dashed lines represent the
estimated locations of structural breaks. As clearly seen in Figure 28, the estimated boundary is not close to the actual boundary. But it is still a reasonable structural change since no
Figure 28: Combined durations for BAC during Wed. June 5 and Thur. June 6, 2013.
Black dashed lines represent the boundaries between different days. Red dashed lines represent the estimated locations of structural breaks. Note that the first red dashed line overlaps with the first black dashed line.
seemingly change in pattern occurs before the first estimated boundary. The extra estimated structural breaks is within Thur. June 6. Although no any related news were reported on that day, dividing the durations of the second day into two clusters (short vs long) are reasonable since the mean of durations before the second red line is relatively smaller than the mean of durations after red line (as seen in Figure 28).