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A NOTE ON INEQUALITY CONSTRAINTS IN THE GARCH MODEL

Published online by Cambridge University Press:  26 February 2008

Henghsiu Tsai*
Affiliation:
Academia Sinica
Kung-Sik Chan
Affiliation:
University of Iowa
*
Address correspondence to Henghsiu Tsai, Institute of Statistical Science, Academia Sinica, 128, Academia Rd. Sec. 2, Taipei 115, Taiwan; e-mail: [email protected].

Abstract

We consider the parameter restrictions that need to be imposed to ensure that the conditional variance process of a GARCH(p,q) model remains nonnegative. Previously, Nelson and Cao (1992, Journal of Business ’ Economic Statistics 10, 229–235) provided a set of necessary and sufficient conditions for the aforementioned nonnegativity property for GARCH(p,q) models with p ≤ 2 and derived a sufficient condition for the general case of GARCH(p,q) models with p ≥ 3. In this paper, we show that the sufficient condition of Nelson and Cao (1992) for p ≥ 3 actually is also a necessary condition. In addition, we point out the linkage between the absolute monotonicity of the generalized autoregressive conditional heteroskedastic (GARCH) generating function and the nonnegativity of the GARCH kernel, and we use it to provide examples of sufficient conditions for this nonnegativity property to hold.

Type
NOTES AND PROBLEMS
Copyright
Copyright © Cambridge University Press 2008

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References

REFERENCES

Bollerslev, T. (1986) Generalized autoregressive conditional heteroscedasticity. Journal of Econometrics 31, 307327.CrossRefGoogle Scholar
Conrad, C.Haag, B. (2006) Inequality constraints in the fractionally integrated GARCH model. Journal of Financial Econometrics 4, 413449.CrossRefGoogle Scholar
Engle, R.F. (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50, 9871008.CrossRefGoogle Scholar
Feller, W. (1968) An Introduction to Probability Theory and Its Applications, vol. 1. Wiley.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and Its Applications, vol. 2. Wiley.Google Scholar
Nelson, D.B.Cao, C.Q. (1992) Inequality constraints in the univariate GARCH model. Journal of Business ’ Economic Statistics 10, 229235.CrossRefGoogle Scholar
Titchmarsh, E.C. (1939) The Theory of Functions, 2nd ed. Oxford University Press.Google Scholar
Tsai, H.Chan, K.S. (2006) A Note on Inequality Constraints in the GARCH Model. Technical Report No. 361, Department of Statistics and Actuarial Science, University of Iowa; downloadable from http://1#x002F;techrep/tr361.pdf.Google Scholar
Tsai, H.Chan, K.S. (2007) A note on non-negative ARMA processes. Journal of Time Series Analysis 28, 350360.CrossRefGoogle Scholar
Widder, D.V. (1946) The Laplace Transform. Princeton University Press.Google Scholar