Several Extended CAViaR Models and Their Applications to the VaR Forecasting of the Security Markets

The conditional autoregressive Value-at-Risk (CAViaR) model, as a conditional autoregressive specification for calculating the Value-at-Risk (VaR) of the security market, has been receiving more and more attentions in recent years. As asymmetry may have a significant influence on the markets and the returns may have an autoregressive mean, this study proposes some extended CAViaR models, including asymmetric indirect threshold autoregressive conditional heteroskedasticity (TARCH) model and indirect generalized autoregressive conditional heteroskedasticity (GARCH) model with an autoregressive mean. We also present two types of CAViaR-Volatility models by adding the volatility term as an exogenous explanatory variable. Our empirical results indicate that extended models perform more effectively on out-of-sample predictions, as both forecasting effect and model stability have been improved. In addition, we find that the forecasting effect is better at the lower quantile (1%) than at the higher quantile (5%); a possible explanation is that extreme market information has more impact on VaR. In addition, there is negative correlation between volatility and VaR; VaR decreases as volatility increases.