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

Abstract

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 µ1 = 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.

Monitoring multivariate variance changes

“…case indicating that the time series of GARCH residuals show similar characteristics as the underlying innovation vectors. A detailed investigation of this behavior that was considered for the univariate case in Kulperger and Yu (2005) is the object of future research.…”

Time Series Segmentation Procedures to Detect, Locate and Estimate Change-Points

“…Other authors were interested in cusums of squares of the residuals of a GARCH model. For instance, Kulperger and Yu (2005) constructed high moment partial sum processes of residuals (from squares until fourth moment) in a GARCH model and provide interesting diagnostic tools.…”

A note on the Jarque–Bera normality test for GARCH innovations

“…See Bai and Ng (2005), and Chen and Kuan (2003). Recently, Kulperger and Yu (2005) studied the JB test based on GARCH residuals within the framework of a high moment partial sum process. However, the JB test and regularity conditions they employ should be updated since more sophisticated techniques have been developed concerning the asymptotic theories for GARCH models.…”