Estimation in functional lagged regression

The paper introduces a functional time series (lagged) regression model. The impulse response coefficients in such a model are operators acting on a separable Hilbert space, which is the function space L2 in applications. A spectral approach to the estimation of these coefficients is proposed and asymptotically justified under a general nonparametric condition on the temporal dependence of the input series. Since the data are infinite dimensional, the estimation involves a spectral domain dimension reduction technique. Consistency of the estimators is established under general data dependent assumptions on the rate of the dimension reduction parameter. Their finite sample performance is evaluated by a simulation study which compares two ad hoc approaches to dimension reduction with a new asymptotically justified method. The new method is superior when the MSE of the in sample prediction error is used as a criterion.