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

• 成果类型: Article
• 署名作者: Kulperger, R; Yu, H
• 署名单位: Western University (University of Western Ontario)
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/009053605000000534
• 发表日期: 2005
• 页码: 2395-2422
• 关键词: normality; stationarity; tests; time; ARCH

摘要

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 kill 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 kill moment mu(k) of the innovation sequence. If mu(k) = 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 kill 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 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.