The usual function-on-function linear regression model depicts the association between functional variables in the whole rectangular region and the value of response curve at any point is influenced by the entire trajectory of the predictor curve. But in addition to this, there are cases where the value of the response curve at a point is only influenced by the value of the predictor curve in a subregion, such as the historical relationship and the short-term association. We will consider the restricted function-on-function regression model, where the value of response curve at any point is influenced by a subtrajectory of the predictor. We have two major purposes. First, we propose a novel estimation procedure that is more accurate and computational efficient for the restricted function-on-function model with a given subregion. Second, as the subregion is seldom specified in practice, we propose a subregion selection procedure that can lead to models with better interpretation and predictive performance. Algorithms are developed for both model estimation and subregion selection.
Keywords: historical function-on-function; optimal expansion; region selection; restricted function-on-function regression model.
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