Functional linear regression with functional response

In this paper, we develop new estimation results for functional regressions where both the regressor Z(t) and the response Y(t) are functions of Hilbert spaces, indexed by the time or a spatial location. The model can be thought as a generalization of the multivariate regression where the regression coefficient is now an unknown operator Pi. We propose to estimate the operator Pi by Tikhonov regularization, which amounts to apply a penalty on the L-2 norm of Pi. We derive the rate of convergence of the mean square error, the asymptotic distribution of the estimator, and develop tests on Pi. As trajectories are often not fully observed, we consider the scenario where the data become more and more frequent (infill asymptotics). We also address the case where Z is endogenous and instrumental variables are used to estimate Pi. An application to the electricity consumption completes the paper.