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 that compares two ad hoc approaches to dimension reduction with an alternative, asymptotically justified method.