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Non-negativity Conditions for the Hyperbolic GARCH Model

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  • Christian Conrad

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 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(∞) 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.

Suggested Citation

  • Christian Conrad, 2007. "Non-negativity Conditions for the Hyperbolic GARCH Model," KOF Working papers 07-162, KOF Swiss Economic Institute, ETH Zurich.
    • Handle: RePEc:kof:wpskof:07-162
    DOI: 10.3929/ethz-a-005390226
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    More about this item

    Keywords

    Inequality constraints; Fractional integration; Long memory GARCH processes;
    All these keywords.

    JEL classification:

    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes
    • C52 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Evaluation, Validation, and Selection
    • C53 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Forecasting and Prediction Models; Simulation Methods

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