Bootstrap Confidence Bands for the Autoregression Function

We derive a strong approximation of a local polynomial estimator (LPE) in nonparametric autoregression by an LPE in a corresponding nonparame-tric regression model. This generally suggests the application of regression-typical tools for statistical inference in nonparametric autoregressive models. It provides an important simpliication for the bootstrap method to be used: It is enough to mimic the structure of a nonparametric regression model rather than to imitate the more complicated process structure in the autoregressive case. As an example we consider a simple wild bootstrap. Besides our particular application to simultaneous conndence bands, this suggests the validity of wild bootstrap for several other statistical purposes. 1. Introduction In this paper we deal with a nonparametric autoregressive model X t = m(X t?1) + " t : Such processes generalize well-known linear rst order autoregressive models. Several authors dealt with the interesting statistical problem of estimating m nonparametri-cally. Robinson (1983), Tjjstheim (1994) and Masry and Tjjstheim (1995) dealt with usual Nadaraya-Watson type estimators. Recently (HH ardle and Tsybakov (1995)) the interest was directed to local polynomial estimators for this setup. Of course, it is important to get knowledge about the statistical properties of particular nonparame-tric estimates. Besides asymptotic results the bootstrap ooers a powerful tool for this purpose. Franke, Kreiss and Mammen (1996) consider a time series speciic bootstrap as well as a wild bootstrap proposal in order to obtain pointwise conndence intervals for kernel smoothers in nonparametric autoregression with conditional heterosceda-sticity. Successful application of the bootstrap for time series models can be found for example in Tjjstheim and Auestad (1994). In this paper we consider the situation from a more general point of view. As a typical nonparametric estimator we consider local polynomials. We derive a strong approximation of a local polynomial estimator (LPE) in the autoregressive setup by an LPE in a corresponding nonparametric regression model. Besides the application of this main result to our particular example of simultaneous conndence bands, it contains the general message that nonparametric autoregression and nonparametric regression are asymptotically equivalent in a certain sense concerning statistical inference about the autoregression/regression function. Of course, this suggests and justiies to use regression-type methods for statistical inference in the context of nonparametric autoregression, too. Further, from Neumann and Polzehl (1995) it is essentially known that one can nd a strong approximation of an LPE in nonparametric regression by a random process generated by an appropriate bootstrap technique. Together …