Where does the tail begin? An approach based on scoring rules

Learning about the tail shape of time series is important in, e.g., economics, finance, and risk management. However, it is well known that estimates of the tail index can be very sensitive to the choice of the number <i>k</i> of tail observations used for estimation. We propose a procedure that determines where the tail begins by choosing <i>k</i> in a data-driven fashion using scoring rules. So far, scoring rules have mainly been used to compare density forecasts. We also demonstrate how our proposal can be used in multivariate applications in the system risk literature. The advantages of our choice of <i>k</i> are illustrated in simulations and an empirical application to Value-at-Risk forecasts for five U.S. blue-chip stocks.