The Annals of Statistics

High moment partial sum processes of residuals in GARCH models and their applications

Reg Kulperger and Hao Yu

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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.

Article information

Ann. Statist., Volume 33, Number 5 (2005), 2395-2422.

First available in Project Euclid: 25 November 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F17: Functional limit theorems; invariance principles 62M99: None of the above, but in this section 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]

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


Kulperger, Reg; Yu, Hao. High moment partial sum processes of residuals in GARCH models and their applications. Ann. Statist. 33 (2005), no. 5, 2395--2422. doi:10.1214/009053605000000534.

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