Non-Negativity Conditions for the Hyperbolic GARCH Model

Journal of Econometrics, Forthcoming

KOF Working Paper No. 162

40 Pages Posted: 17 May 2007 Last revised: 4 May 2010

See all articles by Christian Conrad

Christian Conrad

Heidelberg University - Alfred Weber Institute for Economics; ETH Zürich - KOF Swiss Economic Institute

Date Written: April 1, 2007

Abstract

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) 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(∞) coeffcients 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.

Keywords: Inequality constraints, fractional integration, long memory GARCH processes

JEL Classification: C22, C52, C53

Suggested Citation

Conrad, Christian, Non-Negativity Conditions for the Hyperbolic GARCH Model (April 1, 2007). Journal of Econometrics, Forthcoming, KOF Working Paper No. 162, Available at SSRN: https://ssrn.com/abstract=986594 or http://dx.doi.org/10.2139/ssrn.986594

Christian Conrad (Contact Author)

Heidelberg University - Alfred Weber Institute for Economics ( email )

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HOME PAGE: http://www.uni-heidelberg.de/conrad

ETH Zürich - KOF Swiss Economic Institute ( email )

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