On the Identifiability of the Functional Convolution Model

TL;DR

This paper establishes conditions for the unique identifiability of the functional convolution model's coefficients from OLS estimates, emphasizing the role of the covariate functions' span relative to differential equations.

Contribution

It provides theoretical conditions ensuring the model's identifiability without dimension reduction or smoothing penalties.

Findings

Unique identifiability when covariate functions are not spanned by differential equation solutions.

Functional coefficients are uniquely determined in Sobolev space under specified conditions.

No need for smoothing penalties or dimension reduction for model identification.

Abstract

This report details conditions under which the Functional Convolution Model described in \citet{AHG13} 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.