Model Specification Testing in Nonparametric Time Series Regression with Nonstationarity

This paper considers a class of nonparametric autoregression models with nonstationarity in the mean and then a class of nonparametric time series regression models with nonstationarity in both the conditional mean and conditional variance. For the nonparametric 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 conditional mean of the nonparametric regression is of a known parametric form indexed by a vector of unknown parameters. We then 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 novel 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 how to implement the proposed tests in practice.