Inference on nonparametrically trending time series with fractional errors

The central limit theorem for nonparametric kernel estimates of a smooth trend, with linearly-generated errors, indicates asymptotic independence and homoscedasticity across fixed points, irrespective of whether disturbances have short memory, long memory, or antipersistence. However, the asymptotic variance depends on the kernel function in a way that varies across these three circumstances, and in the latter two involves a double integral that cannot necessarily be evaluated in closed form. For a particular class of kernels, we obtain analytic formulae. We discuss extensions to more general settings, including ones involving possible cross-sectional or spatial dependence.