Proportional functional coefficient time series models

In this paper, we study a new class of semiparametric models, termed as the proportional functional-coefficient linear regression models for time series data. The model can be viewed as a generalization of the functional-coefficient regression models but it has different proportional functions of parameter and different smoothing variables in the same coefficient function in different position. When the parameter is known, the local linear technique is employed to give the initial estimator of the coefficient function in the model, which does not share the optimal rate of convergence. To improve its convergent rate, a one-step backfitting technique is used to obtain the optimal estimator of the coefficient function. The asymptotic properties of the proposed estimators are investigated. When the parameter is unknown, the method of estimating parameter is given. It can be shown that the estimator of the parameter is n -consistent. The bandwidths and the smoothing variables are selected by a data-driven method. A simulated example with two cases and two real data examples are used to illustrate the applications of the model.