Nonparametric Specification Testing in Nonlinear and Nonstationary Time Series Models: Theory and Practice

In this paper, we consider some specification testing problems in nonlinear time series models with nonstationarity. We propose using a nonparametric kernel test for specifying whether the regression function is of a known parametric nonlinear form. The power function of the proposed nonparametric test is systematically studied and an asymptotic distribution of the test statistic is shown to depend on the asymptotic behavior of the so called "distance function" under a sequence of general semiparametric local alternatives. The asymptotic theory developed in this paper differs from existing work on nonparametric specification testing in the stationary time series case. In order to implement the proposed test in practice, a computer-intensive bootstrap simulation procedure is introduced and asymptotic approximations for both the size and power functions are established. Furthermore, the bandwidth involved in the test is selected by maximizing the power function while the size function is controlled by a significance level. Meanwhile, both simulated and real data examples are provided to illustrate the proposed theory and methodology.