Regression models with functional covariates for functional responses constitute a powerful

and increasingly important model class. However, regression with functional data poses challenging

problems of non-identifiability. We describe these identifiability issues in realistic applications of penalized linear

function-on-function-regression and delimit the set of circumstances under which they arise.

Specifically, functional covariates whose empirical covariance has lower effective rank than the number of marginal

basis function used to represent the coefficient surface can lead to unidentifiability. Extensive simulation studies

validate the theoretical insights, explore the extent of the problem and allow us to evaluate its practical

consequences under varying assumptions about the data generating processes. Based on theoretical considerations

and our empirical evaluation, we provide easily verifiable criteria for lack of identifiability

and provide actionable advice for avoiding spurious estimation artifacts.

Applicability of our strategy for mitigating non-identifiability is demonstrated

in a case study on the Canadian Weather data set.