  We investigate the behavior of the Generalized Likelihood Ratio Test (GLRT)
(Fan, Zhang and Zhang [Ann. Statist. 29 (2001) 153-193]) for time varying
coefficient models where the regressors and errors are non-stationary time
series and can be cross correlated. It is found that the GLRT retains the
minimax rate of local alternative detection under weak dependence and
non-stationarity. However, in general, the Wilks phenomenon as well as the
classic residual bootstrap are sensitive to either conditional
heteroscedasticity of the errors, non-stationarity or temporal dependence. An
averaged test is suggested to alleviate the sensitivity of the test to the
choice of bandwidth and is shown to be more powerful than tests based on a
single bandwidth. An alternative wild bootstrap method is proposed and shown to
be consistent when making inference of time varying coefficient models for
non-stationary time series.
