Functional coefficient autoregressive models: Estimation and tests of hypotheses

In this paper we study nonparametric estimation and hypothesis testing procedures for the functional coefficient AR (FAR) models of the form Xt = f1(Xt-d)Xt-1 +…+ fp(Xt-d)Xt-p +εt, first 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 successful parametric nonlinear time series models such as the threshold AR models of Tong (1983), exponential AR models of Haggan and Ozaki (1978) and many others. We propose a local linear estimation procedure for estimating the coefficient functions and study its asymptotic properties. In addition, we propose two testing procedures. The first one tests whether all the coefficient functions are constant (i.e. whether the process is linear). The second one tests if all the coefficient functions are continuous, (i.e. if any threshold type of nonlinearity presents in the process). Some simulation results are presented.