Extreme Conditional Expectile Estimation for Heavy-Tailed ARMA-GARCH Models

Expectiles have recently received considerable attention due to their coherence as a measure of tail risk. Estimating conditional expectiles (CExps), particularly at both intermediate and extreme levels, is crucial in quantitative risk management. This article proposes an ARMA-GARCH model that accommodates fewer finite moments and assumes innovations follow a Pareto-type tail distribution. We apply the two-step self-weighted procedure of He et al. (2022) to forecast extreme CExps. Employing extreme value theory, we estimate the extremal CExp and develop a unified asymptotic theory for the CExp estimator, which incorporates both intermediate and extreme scenarios. Our Monte Carlo simulations demonstrate that the proposed approach significantly improves coverage probabilities compared to other competing methods across various contexts, especially in extreme scenarios. Finally, an empirical application to the daily negative log-returns of fourteen financial asset indices shows that our method consistently outperforms forecasts from Hoga (2022) and the Peaks Over Threshold approach during both normal and crisis periods.