Abstract: We propose a novel functional linear model incorporating latent factors, where scalar response, scalar covariates, and functional covariates have repeated measurements for each subject. Our model accounts for latent factors that may impact the response but remain unobservable. To unveil and estimate these latent factors, we propose an iterated profile estimation method. We then establish the consistency and asymptotic properties of the estimators. To demonstrate the efficacy of our proposed estimation procedure, we conduct simulation studies across various scenarios. We compare our results with estimations derived from conventional functional linear models, revealing the superior performance of our method in addressing latent factors. We further illustrate our proposed model and methodology by analyzing real data from both financial markets and air pollution datasets. In these analyses, we successfully uncover hidden factors that exert influence in these specific fields.

Key words and phrases: Factor model, functional data analysis, functional regression, penalized spline, profile estimation.