Essays on conditional expectiles and extremiles - Oliveira, Víctor Henriques de (2024)
We investigate the behavior of Asymmetric Least Squares (ALS) estimators of conditional risk measures in the context of conditional heteroskedasticity. In particular, we focus on the estimation of the conditional expectile (Newey and Powell, 1987) and extremile (Daouia et al., 2019) in a GARCH framework because they are both law-invariant and coherent risk measures. We assess estimation risk by examining how the first-step GARCH estimation affects the asymptotic behavior of the ALS estimators. In particular, we derived both consistency and asymptotic normality of the two-step estimator for coditional expectiles. Monte Carlo simulations examine the effectiveness of several bootstrap approaches, showing that fixed-design residual resampling entails robust prediction intervals. Our empirical analysis reveals that conditional Asymmetric Least Squares estimators are perfectly reasonable candidates to assess the risk in the same manner as traditional quantile-based risk measures, also offering a complementary view on the dynamics of the gain-loss ratio.