Recently, Cavaliere, Georgiev, and Taylor (2018, Econometric Theory 34, 302–348) (CGT) considered the augmented Dickey–Fuller (ADF) test for a unit-root model with linear noise driven by i.i.d. infinite variance innovations and showed that ordinary least squares (OLS)-based ADF statistics have the same distribution as in Chan and Tran (1989, Econometric Theory 5, 354–362) for i.i.d. infinite variance noise. They also proposed an interesting question to extend their results to the case with infinite variance GARCH innovations as considered in Zhang, Sin, and Ling (2015, Stochastic Processes and their Applications 125, 482–512). This paper addresses this question. In particular, the limit distributions of the ADF for random walk models with short-memory linear noise driven by infinite variance GARCH innovations are studied. We show that when the tail index $\alpha <2$ , the limit distributions are completely different from that of CGT and the estimator of the parameters of the lag terms used in the ADF regression is not consistent. This paper provides a broad treatment of unit-root models with linear GARCH noises, which encompasses the commonly entertained unit-root IGARCH model as a special case.

中文翻译：

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具有无限方差 GARCH 误差的非平稳线性过程

最近，Cavaliere、Georgiev 和 Taylor (2018,计量经济学理论34, 302–348) (CGT) 考虑了对具有由 iid 无限方差创新驱动的线性噪声的单位根模型的增强的 Dickey-Fuller (ADF) 检验，并表明基于普通最小二乘 (OLS) 的 ADF 统计具有相同的Chan 和 Tran (1989,计量经济学理论5, 354–362) 用于 iid 无限方差噪声。他们还提出了一个有趣的问题，将他们的结果扩展到具有无限方差 GARCH 创新的情况，如 Zhang、Sin 和 Ling (2015,随机过程及其应用125、482-512）。本文解决了这个问题。特别是，研究了由无限方差 GARCH 创新驱动的具有短记忆线性噪声的随机游走模型的 ADF 极限分布。我们证明当尾部索引$\阿尔法 <2$，极限分布与 CGT 完全不同，并且 ADF 回归中使用的滞后项参数的估计量不一致。本文提供了对具有线性 GARCH 噪声的单位根模型的广泛处理，其中包括作为特例的普遍接受的单位根 IGARCH 模型。