Consistent nonparametric multiple regression for dependent heterogeneous processes: The fixed design case

Consider the nonparametric regression model Yi(n) = g(xi(n)) + [var epsilon]i(n), i = 1, ..., n, where g is an unknown regression function and assumed to be bounded and real valued on A [subset of] Rp, xi(n)'s are known and fixed design points and [var epsilon]i(n)'s are assumed to be both dependent and non-identically distributed random variables. This paper investigates the asymptotic properties of the general nonparametric regression estimator gn(x) = [Sigma]i = 1n Wni(x) Yi(n), where the weight function Wni(x) is of the form Wni(x) = Wni(x; x1(n), x2(n), ..., xn(n). The estimator gn(x) is shown to be weak, mean square error, and universal consistent under very general conditions on the temporal dependence and heterogeneity of [var epsilon]i(n)'s. Asymptotic distribution of the estimator is also considered.