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. We propose an infi?nite dimensional Nadaraya-Watson type estimator with a bandwidth sequence that shrinks the e¤ects of long lags. We investigate its asymptotic properties in detail under both static and dynamic regressions contexts. 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 as 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 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 in a non-trivial way.