  A partial least squares regression is proposed for estimating the
function-on-function regression model where a functional response and multiple
functional predictors consist of random curves with quadratic and interaction
effects. The direct estimation of a function-on-function regression model is
usually an ill-posed problem. To overcome this difficulty, in practice, the
functional data that belong to the infinite-dimensional space are generally
projected into a finite-dimensional space of basis functions. The
function-on-function regression model is converted to a multivariate regression
model of the basis expansion coefficients. In the estimation phase of the
proposed method, the functional variables are approximated by a
finite-dimensional basis function expansion method. We show that the partial
least squares regression constructed via a functional response, multiple
functional predictors, and quadratic/interaction terms of the functional
predictors is equivalent to the partial least squares regression constructed
using basis expansions of functional variables. From the partial least squares
regression of the basis expansions of functional variables, we provide an
explicit formula for the partial least squares estimate of the coefficient
function of the function-on-function regression model. Because the true forms
of the models are generally unspecified, we propose a forward procedure for
model selection. The finite sample performance of the proposed method is
examined using several Monte Carlo experiments and two empirical data analyses,
and the results were found to compare favorably with an existing method.
