Estimation and tests for power-transformed and threshold GARCH models
Estimation and tests for power-transformed and threshold GARCH models
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DOI：
10.1016/j.jeconom.2007.06.004
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发表时间：
2007-07
影响因子：
6.3
通讯作者：
Jiazhu Pan;Hui Wang;H. Tong
中科院分区：
文献类型：
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作者：
Jiazhu Pan;Hui Wang;H. Tong
考虑一类幂变换门限GARCH(p，q)(PTTGRACH(p，q))模型，它是Hwang和Basawa[2004]中幂变换门限GARCH(1，1)模型的自然推广。Box-Cox变换门限GARCH(1，1)过程的平稳性和矩结构统计与概率通讯68,209-220。]并将标准GARCH模型和许多其他模型作为特例包括在内。在误差分布具有有限四阶矩的条件下，我们首先建立了参数的拟极大似然估计(QMLE)的渐近正态分布。对于重尾误差情形，我们给出了PTTGARCH(p，q)模型的最小绝对偏差估计，并证明了在很弱的矩条件下，最小绝对偏差估计是渐近正态分布的。这为基于渐近正态分布的重尾PTTGARCH(p，q)模型的统计推断铺平了道路。因此，我们可以构造GARCH结构的Wald检验，并讨论重尾情况下的阶数选择问题。数值结果表明，对于重尾误差，LADE比QMLE具有更高的精度。此外，将该理论应用于香港恒生指数的日收益率，表明金融时间序列可能存在非对称性和非线性，PTTGARCH模型能够捕捉到这些特征。对于PTTGARCH(p，q)模型的概率结构，我们在附录中给出了该模型存在严格平稳解、存在矩和严格平稳解的尾部行为的充要条件。
Consider a class of power-transformed and threshold GARCH(p,q) (PTTGRACH(p,q)) model, which is a natural generalization of power-transformed and threshold GARCH(1,1) model in Hwang and Basawa [2004. Stationarity and moment structure for Box–Cox transformed threshold GARCH(1,1) processes. Statistics & Probability Letters 68, 209–220.] and includes the standard GARCH model and many other models as special cases. We first establish the asymptotic normality for quasi-maximum likelihood estimators (QMLE) of the parameters under the condition that the error distribution has finite fourth moment. For the case of heavy-tailed errors, we propose a least absolute deviations estimation (LADE) for PTTGARCH(p,q) model, and prove that the LADE is asymptotically normally distributed under very weak moment conditions. This paves the way for a statistical inference based on asymptotic normality for heavy-tailed PTTGARCH(p,q) models. As a consequence, we can construct the Wald test for GARCH structure and discuss the order selection problem in heavy-tailed cases. Numerical results show that LADE is more accurate than QMLE for heavy-tailed errors. Furthermore, the theory is applied to the daily returns of the Hong Kong Hang Seng Index, which suggests that asymmetry and nonlinearity could be present in the financial time series and the PTTGARCH model is capable of capturing these characteristics. As for the probabilistic structure of PTTGARCH(p,q) model, we give in the appendix a necessary and sufficient condition for the existence of a strictly stationary solution of the model, the existence of the moments and the tail behavior of the strictly stationary solution.