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October 2005 High moment partial sum processes of residuals in GARCH models and their applications
Reg Kulperger, Hao Yu
Ann. Statist. 33(5): 2395-2422 (October 2005). DOI: 10.1214/009053605000000534


In this paper we construct high moment partial sum processes based on residuals of a GARCH model when the mean is known to be 0. We consider partial sums of kth powers of residuals, CUSUM processes and self-normalized partial sum processes. The kth power partial sum process converges to a Brownian process plus a correction term, where the correction term depends on the kth moment μk of the innovation sequence. If μk=0, then the correction term is 0 and, thus, the kth power partial sum process converges weakly to the same Gaussian process as does the kth power partial sum of the i.i.d. innovations sequence. In particular, since μ1=0, this holds for the first moment partial sum process, but fails for the second moment partial sum process. We also consider the CUSUM and the self-normalized processes, that is, standardized by the residual sample variance. These behave as if the residuals were asymptotically i.i.d. We also study the joint distribution of the kth and (k+1)st self-normalized partial sum processes. Applications to change-point problems and goodness-of-fit are considered, in particular, CUSUM statistics for testing GARCH model structure change and the Jarque–Bera omnibus statistic for testing normality of the unobservable innovation distribution of a GARCH model. The use of residuals for constructing a kernel density function estimation of the innovation distribution is discussed.


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Reg Kulperger. Hao Yu. "High moment partial sum processes of residuals in GARCH models and their applications." Ann. Statist. 33 (5) 2395 - 2422, October 2005.


Published: October 2005
First available in Project Euclid: 25 November 2005

zbMATH: 1086.62100
MathSciNet: MR2211090
Digital Object Identifier: 10.1214/009053605000000534

Primary: 60F17 , 62M10 , 62M99

Keywords: CUSUM , GARCH , high moment partial sum process , kurtosis , omnibus , residuals , skewness , sqrt{n} consistency , weak convergence

Rights: Copyright © 2005 Institute of Mathematical Statistics


Vol.33 • No. 5 • October 2005
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