Abstract: We use a reproducing kernel Hilbert space approach to develop a methodology for testing hypotheses about the slope function in a functional linear regression for time series. In contrast to most existing studies, which 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. We propose an asymptotically pivotal test that does not require estimating nuisance parameters or long-run covariances. The key technical tools that we use 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. Lastly, we demonstrate the potential of our methods using a small simulation study and a data example.

Key words and phrases: Functional linear regression, functional time series, m-approximability, relevant hypotheses, reproducing kernel Hilbert space, self-normalization.