NON‐PARAMETRIC ESTIMATION WITH STRONGLY DEPENDENT MULTIVARIATE TIME SERIES

Abstract

Smooth non‐parametric kernel density and regression estimators are studied when the data are strongly dependent. In particular, we derive central (and non‐central) limit theorems for the kernel density estimator of a multivariate Gaussian process and an infinite‐order moving average of an independent identically distributed process, as well as the estimator's consistency for other types of data, such as non‐linear functions of a Gaussian process. We find that the kernel density estimator at two different points, under certain conditions, is not only perfectly correlated but may converge to the same random variable. Also, central (and non‐central) limit theorems of the non‐parametric kernel regression estimator are studied. One important and surprising characteristic found is that its asymptotic variance does not depend on the point at which the regression function is estimated and also that its asymptotic properties are the same whether or not regressors are strongly dependent. Finally, a Monte Carlo experiment is reported to assess the behaviour of the estimators in finite samples.