Specification Test for Threshold Estimation in Extreme Value Theory

A fundamental component in the modeling of a financial risk exposure is the estimation of the probability distribution function that best describes the true data-generation process of independent and extreme loss events that fall above a certain threshold. In this paper, we assume that, above the threshold, the extreme loss events are explained by an extreme value distribution. For that purpose, we apply the classical peaks-over-threshold method in extreme-value statistics. According to that approach, data in excess of a certain threshold is asymptotically described by a generalized Pareto distribution (GPD). Consequently, establishing a mechanism to estimate this threshold is of major importance. The current methods to estimate the thresholds are based on a subjective inspection of mean excess plots or other statistical measures; the Hill estimator, for example, leads to an undesirable level of subjectivity. In this paper, we propose an innovative mechanism that increases the level of objectivity of threshold selection, departing from a subjective and imprecise eyeballing of charts. The proposed algorithm is based on the properties of the generalized Pareto distribution and considers the choice of threshold to be an important modeling decision that can have significant impact on the model outcomes. The algorithm we introduce here is based on the Hausman specification test to determine the threshold, which maintains proper specification so that the other parameters of the distribution can be estimated without compromising the balance between bias and variance. We apply the test to real risk data so that we can obtain a practical example of the improvements the process will bring. Results show that the Hausman test is a valid mechanism for estimating the GPD threshold and can be seen as a relevant enhancement in the objectivity of the entire process.