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| Untersuchte Arbeit: Seite: 90, Zeilen: 8-31 |
Quelle: Herrera Schipp 2009 Seite(n): 213-214, Zeilen: 21-33;1-2 |
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| The problem with the theory outlined in the previous section is that it assumes that the underlying series is independent, which is unrealistic in most of applications. Serial dependence and volatility cluster play an important role in the most applications on returns of financial series, and so exceedances of a high threshold for daily financial return series do not necessarily occur according to a homogeneous Poisson process. Therefore, the direct application of the POT method is nonviable.
However, under weak conditions the POT representation may be applied to the maximum value of each cluster. The problem here is the identification of independent clusters of exceedances over a high threshold. This is because of the fact that the choice of declustering scheme has often an important impact on estimates of cluster characteristics. Possible algorithms are given by the run method, the block method or the interval method (see Beirlant et al. (2004)). In particular the interval estimator introduced by Ferro et. al. Ferro and Segers (2003) propose an automatic declustering scheme that is justified by an asymptotic result for the arrival times between threshold exceedances. The scheme relies on the estimation of extremal index prior to declustering, which can be interpreted as the reciprocal of the mean cluster size. However, this method consists of a two steps procedure and the cluster behaviour of the extremes is lost. By other way, the methodology introduced in this section takes advantage of the structure of the model, thus allowing the existing (dependent) data to be used more effectively. In the early 1970s, Hawkes (1971) introduced a family of what he called "self-exciting" or "mutually exciting" models, which became both pioneering examples of the conditional intensity methodology. The models have been greatly improved and extended by Ogata (1988), whose ETAS model has been successfully used to elucidate the detailed structure of aftershock sequences. |
The problem with the theory outlined in the previous section is that it assumes that the underlying series is independent, which is unrealistic in most applications. Serial dependence and volatility cluster play an important role in most applications on returns of financial series, and so exceedances of a high threshold for daily financial return series do not necessarily occur according to a homogeneous Poisson process. Therefore, the application direct of the POT method is nonviable.
However, under weak conditions the POT representation may be applied to the maximum value of each cluster. The problem here is the identification of independent clusters of exceedances over a high threshold. This is because of the fact that the choice of declustering scheme often has a significant impact on estimates of cluster characteristics. Possible algorithms are given by the run method, the block method or the interval method (see [1]). In particular, the interval estimator introduced by [8] proposes an automatic declustering scheme that is justified by an asymptotic result for the arrival times between threshold exceedances. The scheme relies on the estimation of extremal index prior to declustering, which can be interpreted as the reciprocal of the mean cluster size. However, this method consists of a two step procedure and the cluster behaviour of the extremes is lost. The methodology introduced in this section takes advantage of the structure of the model, thus allowing the existing (dependent) data to be used more efficiently. In the early 1970s, [9] introduced a family of what he called "self-exciting" or "mutually exciting" models, which became both pioneering examples of the conditional intensity methodology. The models have been greatly improved and extended by [18], whose ETAS model has been successfully used to elucidate the detailed structure of aftershock sequences. |
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