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

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

2006-02-15

arxiv.org/abs/math/0602325v1

Annals of Statistics 2005, Vol. 33, No. 5, 2395-2422

Mathematics

Statistics Theory

Published at http://dx.doi.org/10.1214/009053605000000534 in the Annals of Statistics (http://www.imstat.org/aos/) by the Inst

Scientific paper

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 $k$th powers of residuals, CUSUM processes and self-normalized partial sum processes. The $k$th power partial sum process converges to a Brownian process plus a correction term, where the correction term depends on the $k$th moment $\mu_k$ of the innovation sequence. If $\mu_k=0$, then the correction term is 0 and, thus, the $k$th power partial sum process converges weakly to the same Gaussian process as does the $k$th power partial sum of the i.i.d. innovations sequence. In particular, since $\mu_1=0$, 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 $k$th 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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