Functional Coefficient Autoregressive Models: Estimation and Tests of Hypotheses

In this paper, we study nonparametric estimation and hypothesis testing procedures for the functional coef®cient AR (FAR) models of the form Xt ˆ f1(X tÿd)X tÿ1 ‡ ‡ f p(X tÿd)X tÿ p ‡ å t, ®rst proposed by Chen and Tsay (1993). As a direct generalization of the linear AR model, the FAR model is a rich class of models that includes many useful parametric nonlinear time series models such as the threshold AR models of Tong (1983) and exponential AR models of Haggan and Ozaki (1981). We propose a local linear estimation procedure for estimating the coef®cient functions and study its asymptotic properties. In addition, we propose two testing procedures. The ®rst one tests whether all the coef®cient functions are constant, i.e. whether the process is linear. The second one tests if all the coef®cient functions are continuous, i.e. if any threshold type of nonlinearity presents in the process. The results of some simulation studies as well as a real example are presented.