Nonlinear expectile regression with application to Value-at-Risk and expected shortfall estimation

This paper considers nonlinear expectile regression models to estimate conditional expected shortfall (ES) and Value-at-Risk (VaR). In the literature, the asymmetric least squares (ALS) regression method has been widely used to estimate expectile regression models. However, no literatures rigorously investigated the asymptotic properties of the ALS estimates in nonlinear models with heteroscedasticity. Motivated by this aspect, this paper studies the consistency and asymptotic normality of the ALS estimates and conditional VaR and ES in those models. To illustrate, a simulation study and real data analysis are conducted.