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

By R. Kulperger, Hao Yu. 2005

87 citations recorded by the source corpus.

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 μ k of the innovation sequence. If μ 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 kth power partial sum of the i.i.d. innovations sequence. In particular, since μ 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.

Subfields: Finance, Management Science and Operations Research, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/a51cbb4b099b45525fb28b725faf028a0afda1f8

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Citations in the corpus

ECONOMETRIC ANALYSIS OF VOLATILITY COMPONENT MODELS (2014, background): “362 c© Cambridge University Press 2014 Horvath, Kokoszka, and Teyssière (2001), Horvath, Kokoszka, and Zhang (2006), Inclan and Tiao (1994), Kokoszka and Leipus (2000), and Kulperger and Yu (2005), among others.”
Fitting an Error Distribution in Some Heteroscedastic Time Series Models (2006, result): “Some of the results of Kulperger and Yu [18] can be used to test for the error moments but not for the error d.”