Non-negativity Conditions for the Hyperbolic GARCH Model
推导了HYGARCH模型中条件方差非负的充要条件，并给出了ARCH(1)系数的表示以高效预测多步条件方差，最后通过NYSE数据展示了错误拟合FIGARCH模型的影响。
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 < 2 and sufficient for p > 2 and emerge as natural extensions of the inequality constraints derived in Nelson and Cao (1992) 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(1) 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. An empirical application of the HYGARCH(1; d; 1) model to daily NYSE data illustrates the importance of our results.