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

Zbl 0616.62092
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References:

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