Bootstrap tests for simple structures in nonparametric time series regression

This paper concerns statistical tests for simple structures such as parametric models, lower order models and additivity in a general nonparametric autoregression setting. We propose to use a modified L2-distance between the nonparametric estimator of regression function and its counterpart under null hypothesis as our test statistic which delimits the contribution from areas where data are sparse. The asymptotic properties of the test statistic are established, which indicates the test statistic is asymptotically equivalent to a quadratic form of innovations. A regression type resampling scheme (i.e. wild bootstrap) is adapted to estimate the distribution of this quadratic form. Further, we have shown that asymptotically this bootstrap distribution is indeed the distribution of the test statistics under null hypothesis. The proposed methodology has been illustrated by both simulation and application to German stock index data.