High moment partial sum processes of residuals in GARCH models and their applications
| dc.creator | Kulperger, Reg | |
| dc.creator | Yu, Hao | |
| dc.date | 2006-02-15 | |
| dc.date.accessioned | 2026-07-07T08:07:34Z | |
| dc.date.available | 2026-07-07T08:07:34Z | |
| dc.description | 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 $k$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 $μ_k$ of the innovation sequence. If $μ_k=0$, 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 $μ_1=0$, 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 $k$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 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. | |
| dc.description | Published at http://dx.doi.org/10.1214/009053605000000534 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org) | |
| dc.identifier | https://arxiv.org/abs/math/0602325 | |
| dc.identifier | http://arxiv.org/abs/math/0602325 | |
| dc.identifier | Annals of Statistics 2005, Vol. 33, No. 5, 2395-2422 | |
| dc.identifier | doi:10.1214/009053605000000534 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/130974 | |
| dc.subject | Statistics Theory | |
| dc.subject | 60F17, 62M99, 62M10 (Primary) | |
| dc.title | High moment partial sum processes of residuals in GARCH models and their applications | |
| dc.type | text |