Asymptotic properties of a Nadaraya-Watson type estimator for regression functions of infinite order∗

We consider a class of nonparametric time series regression models in which the regressor takes values in a sequence space and the data are stationary and weakly dependent. Technical challenges that hampered theoretical advances in these models include the lack of associated Lebesgue density and diffi culties with regard to the choice of dependence structure of the data generating process in the dynamic regression framework. We propose an infinite dimensional NadarayaWatson type estimator with a bandwidth sequence that shrinks the effects of long lags. We investigate its asymptotic properties in detail under both static and dynamic regressions contexts, aiming to answer the open questions left by Linton and Sancetta (2009). First we show pointwise consistency of the estimator under a set of mild regularity conditions. We establish a CLT for the estimator at a point under stronger conditions as well for a feasibly studentized version of the estimator, thereby allowing pointwise inference to be conducted. We establish the uniform consistency over a compact set of logarithmically increasing dimension. We specify the explicit rates of convergence in terms of the Lambert W function, and show that the optimal rate that balances the upper bound of squared bias and variance is of logarithmic order, the precise rate depending on the smoothness of the regression function and the dependence of the data.