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 andhomoscedasticity across fixed points, irrespective of whether disturbances haveshort memory, long memory, or antipersistence. However, the asymptotic variancedepends on the kernel function in a way that varies across these threecircumstances, and in the latter two involves a double integral that cannotnecessarily be evaluated in closed form. For a particular class of kernels, weobtain analytic formulae. We discuss extensions to more general settings,including ones involving possible cross-sectional or spatial dependence.