ISSN 1440-771X

We obtain uniform consistency results for kernel-weighted sample covariances in a nonstation-ary multiple regression framework that allows for both fixed design and random design coefficient variation. In the fixed design case these nonparametric sample covariances have different uniform convergence rates depending on direction, a result that differs fundamentally from the random de-sign and stationary cases. The uniform convergence rates derived are faster than the corresponding rates in the stationary case and confirm the existence of uniform super-consistency. The modelling framework and convergence rates allow for endogeneity and thus broaden the practical econometric import of these results. As a specific application, we establish uniform consistency of nonpara-metric kernel estimators of the coefficient functions in nonlinear cointegration models with time varying coefficients and provide sharp convergence rates in that case. For the fixed design models, in particular, there are two uniform convergence rates that apply in two different directions, both rates exceeding the usual rate in the stationary case