Normal approximation in regression

We obtain a unform strong approximation for the distribution of a Nadaraya-Watson kernel estimator of a regression function. The approximation is obtained for general multivariate explanatory variables under an algebraic moment condition on the errors. A stronger rate of convergene result for the normal approximation is obtained at the expense of stronger moment conditions. We use the strong approximation results to derive a normal approximation to the distribution of the fitted values from the model.