Joint value-at-risk and expected shortfall regression for location-scale time series models

This article studies the joint value-at-risk (VaR) and expected shortfall (ES) regression for a wide class of location-scale time series models including autoregressive and moving average models with generalized autoregressive conditional heteroscedasticity errors. In contrast to the quasi-maximum likelihood estimation, we estimate the model parameters with the aim of more accurate VaR and ES estimation. Then, we show consistency and asymptotic normality for parameter estimators under weak regularity conditions. Finally, a simulation study and a real data analysis are shown to illustrate our results.