GARCH模型残差的高矩部分和过程及其应用
High moment partial sum processes of residuals in GARCH models and their applications
摘要：
在本文中，我们构造了基于残差的高矩部分和过程的均值为0时，一个灰色模型。我们考虑$k$残差次幂的部分和、非线性过程和自归一化部分和过程。$k$次幂部分和过程收敛于一个布朗过程加上一个修正项，其中修正项依赖于新息序列的$k$次矩$μ_k$.如果$μ_k=0$，则校正项为0，因此，$k$次幂部分和过程弱收敛到与i.i.d.的$k$次幂部分和过程相同的高斯过程。创新序列特别地，由于$μ_1=0$，这对于第一矩部分和处理成立，但是对于第二矩部分和处理不成立。我们还考虑了自归一化过程，即用残差样本方差进行标准化。这些表现为残差渐近独立同分布。我们还研究了$k$ th和$(k+1)$ st自归一化部分和过程的联合分布。应用程序的变点问题和拟合优度被认为是，特别是，用于测试Gestival模型的结构变化和Jarque-Bera综合统计的Gestival模型的不可观测的创新分布的正态性测试的Gestival统计。利用残差构造新息分布的核密度函数估计进行了讨论。
英文摘要：
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.
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