The Tail Behavior due to the Presence of the Risk Premium in AR-GARCH-in-Mean, GARCH-AR, and Double-Autoregressive-in-Mean Models[Stock Returns and Volatility]

We extend the results in Borkovec (2000), Basrak, David, and Mikosch (2002a), Lange (2011), and Francq and Zakoïan (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 GARCH-M 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.