In this article, we discuss the bootstrap as a tool for statistical inference in econometric time series models. Importantly, in the context of testing, properties of the bootstrap under the null (size) as well as under the alternative (power) are discussed. Although properties under the alternative are crucial to ensure the consistency of bootstrap-based tests, it is often the case in the literature that only validity under the null is discussed. We provide new results on bootstrap inference for the class of double-autoregressive (DAR) models. In addition, we review key examples from the bootstrap time series literature in order to emphasize the importance of properly defining and analyzing the bootstrap generating process and associated bootstrap statistics, while also providing an up-to-date review of existing approaches. DAR models are particularly interesting for bootstrap inference: first, standard asymptotic inference is usually difficult to implement due to the presence of nuisance parameters; second, inference involves testing whether one or more parameters are on the boundary of the parameter space; third, even second-order moments may not exist. In most of these cases, the bootstrap is not considered an appropriate tool for inference. Conversely, and taking testing nonstationarity to illustrate, we show that although a standard bootstrap based on unrestricted parameter estimation is invalid, a correct implementation of the bootstrap based on restricted parameter estimation (restricted bootstrap) is first-order valid. That is, it is able to replicate, under the null hypothesis, the correct limiting distribution. Importantly, we also show that the behavior of this bootstrap under the alternative hypothesis may be more involved because of possible lack of finite second-order moments of the bootstrap innovations. This feature makes for some parameter configurations, the restricted bootstrap unable to replicate the null asymptotic distribution when the null is false. We show that this possible drawback can be fixed by using a novel bootstrap in this framework. For this “hybrid bootstrap,” the parameter estimates used to construct the bootstrap data are obtained with the null imposed, while the bootstrap innovations are sampled with replacement from unrestricted residuals. We show that the hybrid bootstrap mimics the correct asymptotic null distribution, irrespective of the null being true or false. Monte Carlo simulations illustrate the behavior of both the restricted and the hybrid bootstrap, and we find that both perform very well even for small sample sizes.

中文翻译：

---

时间序列模型中假设的自举测试入门：应用于双自回归模型

在本文中，我们讨论了 bootstrap 作为计量经济时间序列模型中统计推断的工具。重要的是，在测试的上下文中，讨论了在 null（大小）和替代（功率）下引导程序的属性。尽管备选方案下的属性对于确保基于引导的测试的一致性至关重要，但文献中通常只讨论零下的有效性。我们为双自回归 (DAR) 模型类提供了引导推理的新结果。此外，我们回顾了 bootstrap 时间序列文献中的关键示例，以强调正确定义和分析 bootstrap 生成过程和相关 bootstrap 统计数据的重要性，同时还提供对现有方法的最新回顾。DAR 模型对于引导推理特别有趣：首先，由于存在令人讨厌的参数，标准渐近推理通常难以实现；其次，推理涉及测试一个或多个参数是否在参数空间的边界上；第三，甚至可能不存在二阶矩。在大多数情况下，引导程序不被认为是合适的推理工具。相反，并以测试非平稳性来说明，尽管基于无限制参数估计的标准引导程序是无效的，但基于受限参数估计（受限引导程序）的引导程序的正确实现是一阶有效的。也就是说，它能够在原假设下复制正确的极限分布。重要的，我们还表明，由于可能缺乏引导创新的有限二阶矩，替代假设下的引导行为可能更复杂。这个特性使得一些参数配置，当 null 为 false 时，受限引导程序无法复制 null 渐近分布。我们表明，可以通过在该框架中使用新颖的引导程序来解决这个可能的缺点。对于这种“混合引导”，用于构建引导数据的参数估计是在施加零值的情况下获得的，而引导创新是通过从不受限制的残差中替换进行采样的。我们表明，混合引导程序模拟了正确的渐近零分布，而不管零是真还是假。