The change-point algorithm is applied to the scaled residuals from the pre-processing model. In Section 2.3, we established the better perfor- mance of the algorithm using function φ2, the thresholdqbτset at the 99th
712 = 3×4models are considered: three specifications defined by the possible inclusion
empirical quantile, and a significance level α = 0.05. In particular, the test with these settings is more powerful in detecting the true change- points and this is substantiated also by the three change-points analysis reported in the supplementary appendix. We use these settings through- out, as a different configuration may increase the probability of missing a change-point or finding spurious change-points.
In Figures 2-4, we report the change-points in the upper tail of the in- dustries that exhibit at least one change-point in any of the 39 windows. Similar results are displayed in Figure 5 for the lower tail of the only two industries that present change-points. Given that there is no evidence of change-points in the lower tail, apart for the random occurrence reported in Figure 5, we conclude that lower tail behaviour remains constant over the year. Contrarily, we report evidence of change-points in the upper tail for nine out of 12 industries, signalling a strong seasonal behaviour in the upper tail. As pointed out in Section 2.3, this evidence is due to pure tail changes given that one-tailed change-points can only occur as a consequence of variation in only one of the tails of the innovation distri- bution.
Apart for slight time variations, the change-points are quite stable within industries, while they tend to be different in terms of number and location across industries. We note the presence of a short Winter season and a long Summer season, with the length of these seasons depending on the industry. In particular, Winter seems to be critical in that fluc- tuation in the tail looks frequent during this period, as witnessed by the appearance of a third change-point that splits this season in two intervals in both the Manufacturing and Shops industries.
In a few windows of the Other industry, a single change-point is ob- served. Focusing on the whole spectrum of change-points across the 39 windows of this industry, we can notice that windows with two change- points very close to each other co-exist with windows with a single change- point. This evidence gives support to the hypothesis put forth in Section 2.3 that only one change-point may be detected if the actual season is quite small. Either the power of the test is too low, or the second change- point happens within the 30-day bound defined by the λ parameter in
Equation (2.6).
Once we have identified the change-points, we proceed with the study of the tail behaviour within each interval. Although we filter the obser- vations with a pre-processing model, it is important to verify the level of dependence in the extremes of the estimated residuals as it may be quite different from that which occurs at mean levels. In Figure 25 we report, for the 10 industries presenting at least one change-point in the upper tail, the estimated values of the extremal index θ obtained with the in- terval estimator of Ferro and Segers (2003). Most of the estimates are in the interval (0.8, 1). We argue that the pre-processing model has filtered away much of the dependence and we treat the residuals as independent. In each identified interval, we fit the GP to the exceedances over a threshold q. There exists a huge literature regarding threshold selection in EVT (Scarrott and MacDonald, 2012), nonetheless a full agreement re- garding the right way to proceed has not been achieved. We fit the GP distribution to data over a sequence of thresholds qn, from the 90th up to
the 99th empirical quantile, and qualitatively evaluate the tail behaviour. To assess the goodness of fit, we visually inspect the QQ-plots. Given the large number of plots, we do not report them here, but note that most of the time the observations nicely adhere to the theoretical lines of the GP. We also test the null hypothesis of a GP distribution with the Anderson-Darling and the Cramer-von Mises tests of Choulakian and Stephens (2001). In Figures 6-14, we report the frequency of rejection and non rejection for both tests at the different qnthresholds.
Specifically, to clearly distinguish between Summer and Winter, we consider only windows exhibiting two seasons. The figures show that in most cases the assumption that the upper tail follows a GP distribution is not rejected. Summer seasons tend to present a higher rate of rejection as a consequence of the larger number of observations. This implies that some of the thresholds are too low for the convergence result of Pickands (1975). On a few occasions, the number of observations over the 99th quantile in Winter is insufficient to compute MLE.
In Figure 15, we show a scatter plot of the ν and ξ parameter esti- mates, obtained over the 10 different threshold levels in qn, for the Sum-
mer and the Winter seasons. Specifically, we discard the estimated values for which the GP distribution null hypothesis was rejected at the 5% level either by the Anderson-Darling or the Cramer-von Mises test. The figure shows that the two seasons are not substantially different with respect to ξ, but they are strongly different in terms of scale. The values of ν are much higher in Winter than in Summer. In Figures 16-24 we report the empirical quantiles ofbηt, considering only those windows for which ex- actly two seasons are identified. The heat map tracks the change in the value of the quantile by changing the colour of the corresponding tile. From these plots two main messages can be extracted. First, the fact that much of the area in the higher quantiles of the Winter plots is covered by red tiles confirms that Winter is subject to more extreme observations. Second, the fact that the difference between the Summer and the Winter plots at the 90th quantile is small confirms that this behaviour is a con- sequence of a pure shift in the tail shape, and not a consequence of the mean or the variance process. Indeed, we compute the average differ- ence between the quantiles at the same level in Winter and Summer, and find that it increases as we move toward the 99th quantile. We conclude that while both seasons present a similar upper tail decay, the magnitude of the tail events is higher in Winter.
Figure 2: Upper tail change-points. Each line corresponds to one of the 39 rolling windows. Lines 1 and 39 represent respectively the time intervals 1926 − 1975and 1964 − 2013, with the other lines representing the intervals ranging in between at yearly steps. The red crosses represents the identified change-points over the year.
Figure 3: Upper tail change-points. Each line corresponds to one of the 39 rolling windows. Lines 1 and 39 represent respectively the time intervals 1926 − 1975and 1964 − 2013, with the other lines representing the intervals ranging in between at yearly steps. The red crosses represents the identified change-points over the year.
Figure 4: Upper tail change-points. Each line corresponds to one of the 39 rolling windows. Lines 1 and 39 represent respectively the time intervals 1926 − 1975and 1964 − 2013, with the other lines representing the intervals ranging in between at yearly steps. The red crosses represents the identified change-points over the year.
Figure 5: Lower tail change-points. Each line corresponds to one of the 39 rolling windows. Lines 1 and 39 represent respectively the time intervals 1926 − 1975and 1964 − 2013, with the other lines representing the intervals ranging in between at yearly steps. The red crosses represents the identified change-points over the year.
Figure 6: Goodness of fit - Business Equipment. Non rejection (light) and rejection (dark) frequencies for the GP null hypothesis. Only windows for which two seasons were identified are considered. The horizontal axis refers to the sequence of threshold levels qn.
Figure 7: Goodness of fit - Chemicals. Non rejection (light) and rejection (dark) frequencies for the GP null hypothesis. Only windows for which two seasons were identified are considered. The horizontal axis refers to the sequence of threshold levels qn. At the 99th quantile of the Winter season it
was not always possible to obtain estimates of the GPD parameters because of the small number of observations.
Figure 8: Goodness of fit - Durable. Non rejection (light) and rejection (dark) frequencies for the GP null hypothesis. Only windows for which two seasons were identified are considered. The horizontal axis refers to the sequence of threshold levels qn.
Figure 9: Goodness of fit - Health.Non rejection (light) and rejection (dark) frequencies for the GP null hypothesis. Only windows for which two sea- sons were identified are considered. The horizontal axis refers to the se- quence of threshold levels qn.
Figure 10: Goodness of fit - Manufacturing.Non rejection (light) and rejec- tion (dark) frequencies for the GP null hypothesis. Only windows for which two seasons were identified are considered. The horizontal axis refers to the sequence of threshold levels qn.
Figure 11: Goodness of fit - Money.Non rejection (light) and rejection (dark) frequencies for the GP null hypothesis. Only windows for which two sea- sons were identified are considered. The horizontal axis refers to the se- quence of threshold levels qn.
Figure 12: Goodness of fit - No Durable.Non rejection (light) and rejection (dark) frequencies for the GP null hypothesis. Only windows for which two seasons were identified are considered. The horizontal axis refers to the sequence of threshold levels qn.
Figure 13: Goodness of fit - Shops.Non rejection (light) and rejection (dark) frequencies for the GP null hypothesis. Only windows for which two sea- sons were identified are considered. The horizontal axis refers to the se- quence of threshold levels qn.
Figure 14: Goodness of fit - Other.Non rejection (light) and rejection (dark) frequencies for the GP null hypothesis. Only windows for which two sea- sons were identified are considered. The horizontal axis refers to the se- quence of threshold levels qn. At the 99th quantile of the Winter season it
was not always possible to obtain estimates of the GPD parameters because of the small number of observations.
Figure 15: GP parameters scatterplot.Estimated parameters of the GP dis- tribution in Equation (2.4). For each industry we consider only windows for which two change-points were identified and plot the estimates for the Winter season (blue) and the Summer season (red).
Figure 16: Empirical tails - Business Equipment. 90th to 99th empirical quantiles for the Summer and Winter seasons. Only the windows with ex- actly two change-points are considered. The tile corresponding to a fixed window and threshold shifts from green to red as the value of the empirical quantile increases.
Figure 17: Empirical tails - Chemicals. 90th to 99th empirical quantiles for the Summer and Winter seasons. Only the windows with exactly two change-points are considered. The tile corresponding to a fixed window and threshold shifts from green to red as the value of the empirical quantile increases.
Figure 18: Empirical tails - Durable.90th to 99th empirical quantiles for the Summer and Winter seasons. Only the windows with exactly two change- points are considered. The tile corresponding to a fixed window and thresh- old shifts from green to red as the value of the empirical quantile increases.
Figure 19: Empirical tails - Health.90th to 99th empirical quantiles for the Summer and Winter seasons. Only the windows with exactly two change- points are considered. The tile corresponding to a fixed window and thresh- old shifts from green to red as the value of the empirical quantile increases.
Figure 20: Empirical tails - Manufacturing.90th to 99th empirical quantiles for the Summer and Winter seasons. Only the windows with exactly two change-points are considered. The tile corresponding to a fixed window and threshold shifts from green to red as the value of the empirical quantile increases.
Figure 21: Empirical tails - Money.90th to 99th empirical quantiles for the Summer and Winter seasons. Only the windows with exactly two change- points are considered. The tile corresponding to a fixed window and thresh- old shifts from green to red as the value of the empirical quantile increases.
Figure 22: Empirical tails - No Durable. 90th to 99th empirical quantiles for the Summer and Winter seasons. Only the windows with exactly two change-points are considered. The tile corresponding to a fixed window and threshold shifts from green to red as the value of the empirical quantile increases.
Figure 23: Empirical tails - Shops.90th to 99th empirical quantiles for the Summer and Winter seasons. Only the windows with exactly two change- points are considered. The tile corresponding to a fixed window and thresh- old shifts from green to red as the value of the empirical quantile increases.
Figure 24: Empirical tails - Other. 90th to 99th empirical quantiles for the Summer and Winter seasons. Only the windows with exactly two change- points are considered. The tile corresponding to a fixed window and thresh- old shifts from green to red as the value of the empirical quantile increases.
Figure 25: Extremal index estimates. Only sectors presenting at least one change-point in any of the 39 windows are reported. The box-plots pool together the tail index estimates over ten thresholds (from the 90th to the 99th quantiles) of both the Summer and the Winter seasons.