Regression-type Inference in Nonparametric Autoregression

1 1 1. Introduction Autoregressive models form an important class of processes in time series analysis. A nonparametric version of these models was introduced by Jones (1978). To allow for heteroscedastic modelling of the innovations, people often consider the model where the " t are assumed to be i.i.d. with mean 0 and variance 1. Several authors dealt with the interesting statistical problem of estimating the autoregression function m nonparametrically. investigated local polynomial estimators in this context. For some particular purposes of statistical inference like the construction of conndence sets and tests of hypotheses, it is also important to get knowledge about the statistical properties of the underlying estimator. Franke, Kreiss and Mammen (1996) consider time-series speciic as well as regression-type bootstrap methods for model (1.1), and showed their consistency for the pointwise behaviour of kernel smoothers of m. One of our goals is to show the validity of one of these bootstrap methods for statistics which concern the joint distribution of nonparametric estimators. This is motivated by potential applications to simultaneous conndence bands and nonparametric tests. In this paper, we also try to consider the situation from a more general point of view. We show rst the closeness, in an appropriate sense, of a model like (1.1) to a corresponding regression model. To simplify notation, we restrict ourselves to the case of one lag, that is p = q = 1. Without additional eeort, we may allow the whole distribution of " t to depend on X t?1. Accordingly, our basic assumption is that X 0 ; : : : ; X T is a realization from a strictly stationary time-homogeneous Markov chain. The validity of our regression-type bootstrap is based on a strong approximation of