The Tail Behavior due to the Presence of the Risk Premium in AR-GARCH-in-Mean, GARCH-AR, and Double-Autoregressivein-Mean Models

We extend the results in Borkovec (2000), Basrak, David, and Mikosch (2002a), Lange (2011), and Francq and Zakoian (2015) by describing the tail behavior when a risk premium component is added in the mean equation of different conditional heteroskedastic processes. We study three types of parametric models: the traditional generalized autoregressive conditional heteroskedastic (GARCH)-M model, the double autoregressive (AR) model with risk premium, and the GARCH-AR model. We find that if an AR process is introduced in the mean equation of a traditional GARCH-M process, the tail behavior is the same as if it is not introduced. However, if we add a risk premium component to the double AR model, then the tail behavior changes with respect to the GARCH-M. The GARCH-AR model also has a different tail index than the traditional AR-GARCH model. In a simulation study, we show that larger tail indexes are associated with the traditional GARCHM model. When the size of the risk premium component increases, the tail index tends to fall. The only exception to this rule occurs in the double AR model when the risk premium depends on log-volatility. Parameter configurations where the strong stationarity condition of the risk premium models fails are also illustrated and discussed.

Automated summary

This paper extends previous studies by investigating the tail behavior when a risk premium component is added in the mean equation of different conditional heteroskedastic processes. Three types of parametric models are studied: the traditional GARCH-M model, the double AR model with risk premium, and the GARCH-AR model. The findings suggest that the introduction of an AR process in the mean equation of a traditional GARCH-M process does not impact the tail behavior, but adding a risk premium component in the double AR model changes the tail behavior compared to the GARCH-M model. Additionally, the GARCH-AR model exhibits a different tail index than the traditional AR-GARCH model. Simulation results show that larger tail indexes are associated with the traditional GARCHM model, and the increase in the size of the risk premium component tends to decrease the tail index, except in the case of the double AR model where the risk premium depends on log-volatility. Illustrations and discussions are provided on parameter configurations where the strong stationarity condition of the risk premium models fails.