In this paper we investigate to what extent the bootstrap can be applied to conditional mean models, such as regression or time series models, when the volatility of the innovations is random and possibly non-stationary. In fact, the volatility of many economic and financial time series displays persistent changes and possible non-stationarity. However, the theory of the bootstrap for such models has focused on deterministic changes of the unconditional variance and little is known about the performance and the validity of the bootstrap when the volatility is driven by a non-stationary stochastic process. This includes near-integrated exogenous volatility processes as well as near-integrated GARCH processes, where the conditional variance has a diffusion limit; a further important example is the case where volatility exhibits infrequent jumps. This paper fills this gap in the literature by developing conditions for bootstrap validity in time series and regression models with non-stationary, stochastic volatility. We show that in such cases the distribution of bootstrap statistics (conditional on the data) is random in the limit. Consequently, the conventional approaches to proofs of bootstrap consistency, based on the notion of weak convergence in probability of the bootstrap statistic, fail to deliver the required validity results. Instead, we use the concept of ‘weak convergence in distribution’ to develop and establish novel conditions for validity of the wild bootstrap, conditional on the volatility process. We apply our results to several testing problems in the presence of non-stationary stochastic volatility, including testing in a location model, testing for structural change using CUSUM-type functionals, and testing for a unit root in autoregressive models. Importantly, we work under sufficient conditions for bootstrap validity that include the absence of statistical leverage effects, i.e., correlation between the error process and its future conditional variance. The results of the paper are illustrated using Monte Carlo simulations, which indicate that a wild bootstrap approach leads to size control even in small samples.

在本文中，我们研究了当创新的波动性是随机的并且可能是非平稳的时，自举可以在多大程度上应用于条件均值模型，例如回归或时间序列模型。事实上，许多经济和金融时间序列的波动性表现出持续变化和可能的非平稳性。然而，此类模型的自举理论侧重于无条件方差的确定性变化，而当波动率由非平稳随机过程驱动时，自举的性能和有效性知之甚少。这包括近积分外生波动率过程以及近积分 GARCH 过程，其中条件方差具有扩散限制；另一个重要的例子是波动率不经常出现跳跃的情况。本文通过开发具有非平稳、随机波动性的时间序列和回归模型中引导有效性的条件，填补了文献中的这一空白。我们表明，在这种情况下，引导统计的分布（以数据为条件）在限制范围内是随机的。因此，基于自举统计概率弱收敛概念的自举一致性证明的传统方法无法提供所需的有效性结果。相反，我们使用“分布中的弱收敛”的概念来开发和建立以波动过程为条件的野生引导有效性的新条件。我们将我们的结果应用于存在非平稳随机波动的几个测试问题，包括在位置模型中进行测试，使用 CUSUM 类型的泛函测试结构变化，并在自回归模型中测试单位根。重要的是，我们在自举有效性的充分条件下工作，包括没有统计杠杆效应，即错误过程与其未来条件方差之间的相关性。论文的结果使用蒙特卡罗模拟进行了说明，这表明即使在小样本中，狂野的自举方法也能实现大小控制。