Non-negativity conditions for the hyperbolic GARCH model

In this article we derive conditions which ensure the non-negativity of the conditional variance in the Hyperbolic GARCH( p , d , q ) (HYGARCH) model of Davidson (2004). The conditions are necessary and sufficient for p = 1 and sufficient for p ≥ 2 and emerge as natural extensions of the inequality constraints derived in Nelson and Cao (1992) and Tsai and Chan (2008) for the GARCH model and in Conrad and Haag (2006) for the FIGARCH model. As a by-product we obtain a representation of the ARCH( ∞ ) coefficients which allows computationally efficient multi-step-ahead forecasting of the conditional variance of a HYGARCH process. We also relate the necessary and sufficient parameter set of the HYGARCH to the necessary and sufficient parameter sets of its GARCH and FIGARCH components. Finally, we analyze the effects of erroneously fitting a FIGARCH model to a data sample which was truly generated by a HYGARCH process. Empirical applications of the HYGARCH( 1 , d , 1 ) model to daily NYSE and DAX30 data illustrate the importance of our results.