Unified Estimation of Conditional Distortion Risk Measures and Expectiles in APARCH‐X Models

ABSTRACT This paper develops a unified asymptotic framework for estimating conditional distortion risk measures and conditional expectiles within the APARCH‐X model. The proposed framework extends that of Hoga ( International Journal of Forecasting 37, 675‐686, 2021), which focuses exclusively on extreme tail levels, by allowing for both extreme and intermediate tail regimes through a single asymptotic condition, , where is the risk level and corresponds to the extreme case and to the intermediate case. Building on extreme value theory, we derive the asymptotic distributions of the proposed estimators and show that the limiting variances differ structurally across the two regimes. The extreme case studied in Hoga (2021) is thereby recovered as a special instance. In addition, we establish a uniform limit theory over a continuum of tail levels, which facilitates the construction of confidence bands for conditional tail risk measures. Empirically, we apply the proposed methodology to daily returns of four major equity indices and incorporate Chicago Board Options Exchange (CBOE) volatility indices as exogenous variables. The results show that including these volatility measures leads to statistically significant improvements in forecasting accuracy for conditional tail risk measures, particularly during crisis periods, as evidenced by Diebold–Mariano tests. Moreover, the model confidence set procedure consistently favours the APARCH‐X specification over conventional alternatives, including APARCH, AR‐GARCH and GARCH models.