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Tail Index Estimation: Quantile-Driven Threshold Selection

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The most extreme events, such as economic crises, are rare but often have a great impact. It is difficult to precisely determine the likelihood of such events because the sample is small. A common statistical technique is to average the logarithmic distance between a threshold, the least extreme of all the extreme events, and extreme events, i.e., Hill’s estimator. The choice of the threshold is important for modelling these events accurately.

We develop a new method for determining the optimal threshold by using various thresholds as an input to model the tail. For the various thresholds, we document the largest difference between the extreme observations a model produces and the extreme observations in the sample. The threshold that produces the smallest difference is the optimal threshold for modelling these tail events. In simulation exercises, we show that this new method outperforms various existing methods.

To demonstrate the economic relevance of choosing the proper threshold, we use daily stock return data from the Center for Research in Security Prices. We show that the various methods produce large differences in estimating the likelihood of these tail events. Furthermore, we show that using a poorly chosen threshold can change the empirical results of previous research.