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

Forecasting extreme negative returns in gold and silver: A discrete‐duration approach to POT models

“…This model scoring rule was primarily designed to assess the accuracy of density forecasts. Hoga 46 proposed QCRPS as a tool for determining where “the tail begins,” so that the threshold to be chosen should result in the best in‐sample estimates of extreme quantiles. In line with this approach, we aim to evaluate and compare the accuracy of in‐sample conditional quantile estimates derived at different values of $$$ u $$$.…”

“…In line with this approach, we aim to evaluate and compare the accuracy of in‐sample conditional quantile estimates derived at different values of $$$ u $$$. However, as opposed to what Hoga 46 proposed, we do not use the Weissman estimator 47 for extreme quantiles, but our quantile estimates are derived from the LACD‐POT models and represent the VaR (see Equation (7) for the ELACD‐POT, WLACD‐POT, BLACD‐POT, and BSLACD‐POT model or Equation (26) for the DWLACD‐POT model). Consequently, our objective is to select a threshold $$$ u $$$ value that produces the best LACD‐POT VaR estimates for a range of sufficiently small coverage rates $$$ \tau $$$.…”

“…For each value $$$ u $$$ and the corresponding LACD‐POT parameter estimates, we then compute the associated VaR levels at a range of different $$$ \tau $$$ values. From a methodological viewpoint, 46 the QCRPS‐based rule for setting the best $$$ u $$$ threshold corresponds to finding a $$$ u $$$ value that minimizes the score function: $$$$ {S}_{QCRPS}^u=\frac{1}{T}\sum \limits_{t=1}^T\frac{1}{1-{\overline{\tau}}^{\ast }}{\int}_{{\overline{\tau}}^{\ast}}^12\left({I}_{\left\{{y}_t\le Va{R}_{\overline{\tau}}^t(u)\right\}}-\overline{\tau}\right)\left( Va{R}_{\overline{\tau}}^t(u)-{y}_t\right)\kern2.56804pt \mathrm{d}\overline{\tau }, $$$$ where $$$ T $$$ is the number of in‐sample observations, $$$ \overline{\tau}=1-\tau $$$, …”

“…For each value u and the corresponding LACD-POT parameter estimates, we then compute the associated VaR levels at a range of different 𝜏 values. From a methodological viewpoint, 46 the QCRPS-based rule for setting the best u threshold corresponds to finding a u value that minimizes the score function:…”