We obtain uniform consistency results for kernel-weighted sample covariances in a nonstationary 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 design 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 nonparametric 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.
Li, Degui; Phillips, Peter C.B.; and Gao, Jiti, "Uniform Consistency of Nonstationary Kernel-Weighted Sample Covariances for Nonparametric Regression" (2013). Cowles Foundation Discussion Papers. 2326.