A NOTE ON INEQUALITY CONSTRAINTS IN THE GARCH MODEL

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+