  A new single-index model that reflects the time-dynamic effects of the single
index is proposed for longitudinal and functional response data, possibly
measured with errors, for both longitudinal and time-invariant covariates. With
appropriate initial estimates of the parametric index, the proposed estimator
is shown to be $\sqrt{n}$-consistent and asymptotically normally distributed.
We also address the nonparametric estimation of regression functions and
provide estimates with optimal convergence rates. One advantage of the new
approach is that the same bandwidth is used to estimate both the nonparametric
mean function and the parameter in the index. The finite-sample performance for
the proposed procedure is studied numerically.
