Pivotal Inference for Function-on-Function Linear Regression via Self-Normalization

We propose a reproducing kernel Hilbert space approach for testing relevant hypotheses regarding the slope function in function-on-function linear regression for time series. In contrast to exact nullity of the slope function, relevant hypotheses refer to a null hypothesis that the slope function vanishes only approximately and deviations from nullity are measured with respect to the \(L^2\) -norm. An asymptotically pivotal test is proposed, which does not require the estimation of nuisance components of the model that is difficult to estimate in practice. A uniform Bahadur representation and a weak invariance principle for a sequential process of estimates of the slope function are derived in order to ensure the validity of our approach. We also develop an implementation of our approach and illustrate the potential of the new methodology by means of a simulation study.