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High moment partial sum processes of residuals in GARCH models and their applications. (English) Zbl 1086.62100

Summary: 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 C. M. Jarque and A. K. Bera omnibus statistic [Int. Stat. Rev. 55, 163–172 (1987; Zbl 0616.62092)] 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.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
60F17 Functional limit theorems; invariance principles
60F05 Central limit and other weak theorems
62M99 Inference from stochastic processes
62M07 Non-Markovian processes: hypothesis testing
62G07 Density estimation

Citations:

Zbl 0616.62092
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References:

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