On the Identifiability of the Functional Convolution Model

This report details conditions under which the Functional Convolution Model described in Asencio et al. (2013) can be identified from Ordinary Least Squares estimates without either dimension reduction or smoothing penalties. We demonstrate that if the covariate functions are not spanned by the space of solutions to linear differential equations, the functional coefficients in the model are uniquely determined in the Sobolev space of functions with absolutely continuous second derivatives. Asencio et al. (2013) introduced the Functional Convolution Model (FCM) in which for each observation i = 1, . . . , n a functional response Yi(t) depends on the short-term history of one or more functional covariates Xij(t), j = 1, . . . , p which are measured on the same time domain t ∈ [0, Ti] along with scalars zik for k = 1, . . . , d. This is expressed mathematically