Conditional Extreme Value Estimation for Dependent Time Series
Martin Bladt, Laurits Glargaard, Theodor Henningsen
TL;DR
This work develops conditional extreme value methods for dependent, heavy-tailed time series with covariates by formulating a functional tail process. It proves consistency under $\alpha$-mixing and asymptotic normality under $\beta$-mixing with broad second-order conditions, using a kernel-based conditional tail estimator and a conditional Hill estimator tied through the identity $\gamma(x)=\int_{1}^{\infty} T^{x}(s)/s\,ds$. The authors provide finite-sample guidance, including bandwidth selection, and validate the approach via simulations under various dependence structures and a real oil-market application, where tail behavior varies with conditioning values like the OVX volatility index. The results yield simple, plug-in variance expressions and offer a practical framework for conditional tail risk assessment, with potential extensions to conditional risk measures and lagged covariate models.
Abstract
We study the consistency and weak convergence of the conditional tail function and conditional Hill estimators under broad dependence assumptions for a heavy-tailed response sequence and a covariate sequence. Consistency is established under $α$-mixing, while asymptotic normality follows from $β$-mixing and second-order conditions. A key aspect of our approach is its versatile functional formulation in terms of the conditional tail process. Simulations demonstrate its performance across dependence scenarios. We apply our method to extreme event modelling in the oil industry, revealing distinct tail behaviours under varying conditioning values.