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Necessary and sufficient moment conditions for the $$\text{GARCH}((r,s)$$ and asymmetric power $$\text{GARCH}((r,s)$$ models. (English) Zbl 1110.62332
Summary: Although econometricians have been using T. Bollerslev ’s [J. Econ. 31, 307–327 (1986; Zbl 0616.62119)] $$\text{GARCH}(r, s)$$ model for over a decade, the higher order moment structure of the model remains unresolved. A sufficient condition for the existence of the higher order moments of the $$\text{GARCH}(r, s)$$ model was given by S. Ling [ J. Appl. Probab. 36, No. 3, 688–705 (1999; Zbl 1109.62340)]. This paper shows that Ling’s condition is also necessary. As an extension, the necessary and sufficient moment conditions are established for Ding, Granger and Engle ’s [J. Empirical Finance 1, 83–106 (1993)] asymmetric power $$\text{GARCH}(r, s)$$ model.

##### MSC:
 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62P20 Applications of statistics to economics
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