  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.
