Necessary and Sufficient Restrictions for Existence of a Unique Fourth Moment of a Univariate Garch(P,Q) Process

25 Pages Posted: 8 Aug 2005

See all articles by Peter A. Zadrozny

Peter A. Zadrozny

U.S. Bureau of Labor Statistics - Department of Labor; CESifo (Center for Economic Studies and Ifo Institute)

Date Written: July 2005

Abstract

A univariate GARCH(p,q) process is quickly transformed to a univariate autoregressive moving-average process in squares of an underlying variable. For positive integer m, eigenvalue restrictions have been proposed as necessary and sufficient restrictions for existence of a unique mth moment of the output of a univariate GARCH process or, equivalently, the 2mth moment of the underlying variable. However, proofs in the literature that an eigenvalue restriction is necessary and sufficient for existence of unique 4th or higher even moments of the underlying variable, are either incorrect, incomplete, or unecessarily long. Thus, the paper contains a short and general proof that an eigenvalue restriction is necessary and sufficient for existence of a unique 4th moment of the underlying variable of a univariate GARCH process. The paper also derives an expression for computing the 4th moment in terms of the GARCH parameters, which immediately implies a necessary and sufficient inequality restriction for existence of the 4th moment. Because the inequality restriction is easily computed in a finite number of basic arithmetic operations on the GARCH parameters and does not require computing eigenvalues, it provides an easy means for computing by hand the 4th moment and for checking its existence for low-dimensional GARCH processes. Finally, the paper illustrates the computations with some GARCH (1,1) processes reported in the literature.

Keywords: state-space form, Lyapunov equations, nonnegative and irreducible matrices

JEL Classification: C32, C63, G12

Suggested Citation

Zadrozny, Peter A., Necessary and Sufficient Restrictions for Existence of a Unique Fourth Moment of a Univariate Garch(P,Q) Process (July 2005). CESifo Working Paper Series No. 1505. Available at SSRN: https://ssrn.com/abstract=773984

Peter A. Zadrozny (Contact Author)

U.S. Bureau of Labor Statistics - Department of Labor ( email )

2 Massachusetts Avenue, NE
Washington, DC 20212
United States

CESifo (Center for Economic Studies and Ifo Institute)

Poschinger Str. 5
Munich, DE-81679
Germany

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