  We study additive function-on-function regression where the mean response at
a particular time point depends on the time point itself as well as the entire
covariate trajectory. We develop a computationally efficient estimation
methodology based on a novel combination of spline bases with an eigenbasis to
represent the trivariate kernel function. We discuss prediction of a new
response trajectory, propose an inference procedure that accounts for total
variability in the predicted response curves, and construct pointwise
prediction intervals. The estimation/inferential procedure accommodates
realistic scenarios such as correlated error structure as well as sparse and/or
irregular designs. We investigate our methodology in finite sample size through
simulations and two real data applications.
