We develop methodology for testing hypotheses regarding the slope function in functional linear regression for time series via a reproducing kernel Hilbert space approach. In contrast to most of the literature, which considers tests for the exact nullity of the slope function, we are interested in the null hypothesis that the slope function vanishes only approximately, where deviations are measured with respect to the $L^2$-norm. An asymptotically pivotal test is proposed, which does not require the estimation of nuisance parameters and long-run covariances. The key technical tools to prove the validity of our approach include a uniform Bahadur representation and a weak invariance principle for a sequential process of estimates of the slope function. Both scalar-on-function and function-on-function linear regression are considered and finite-sample methods for implementing our methodology are provided. We also illustrate the potential of our methods by means of a small simulation study and a data example.

翻译：我们通过复制内核Hilbert空间方法,为时间序列功能线性回归函数的斜坡函数制定测试方法,与大多数研究斜坡函数完全无效测试的文献不同,我们感兴趣的是空假设,即斜坡函数仅基本消失,对以美元为基值的偏差进行测量。我们建议进行一个非现时关键测试,不需要估计扰动参数和长期的常态。证明我们方法有效性的关键技术工具包括统一的巴哈杜尔代表制和对斜坡函数的测算顺序过程的微弱变化原则。考虑的是斜坡函数和功能对功能的直线回归,并提供执行我们方法的有限方法样本。我们还通过小型模拟研究和数据实例来说明我们方法的潜力。