  Varying-coefficient functional linear models consider the relationship
between a response and a predictor, where the response depends not only the
predictor but also an exogenous variable. It then accounts for the relation of
the predictors and the response varying with the exogenous variable. We
consider the method for truncated estimation for varying-coefficient models.
The proposed method can clarify to what time point the functional predictor
relates to the response at any value of the exogenous variable by investigating
the coefficient function. To obtain the truncated model, we apply the penalized
likelihood method with the sparsity inducing penalty. Simulation studies are
conducted to investigate the effectiveness of the proposed method. We also
report the application of the proposed method to the analysis of crop yield
data to investigate when an environmental factor relates to the crop yield at
any season.
