New Specification Tests in Nonlinear Time Series with Nonstationarity

This paper considers a class of nonparametric autoregressive processes and then a class of nonparametric time series regression models with a nonstationary regressor. For the autoregression case, we propose a nonparametric unit–root test for the conditional mean. For the nonparametric time series regression case, we construct a nonparametric test for testing whether the regression is of a known parametric form indexed by a vector of unknown parameters. We establish asymptotic distributions of the proposed test statistics. Both the setting and the results differ from earlier work on nonparametric time series regression with stationarity. In addition, we develop a bootstrap simulation scheme for the selection of suitable bandwidth parameters involved in the kernel tests as well as the choice of simulated critical values. An example of implementation is given to show that the proposed tests work in practice.