Nonparametric estimation of a smooth trend in the presence of a periodic sequence

Abstract

We investigate the estimators obtained from reversing the three-step estimation procedure of Vogt and Linton (2014) for a regression model including a smooth trend function, a periodic component and a stochastic error term. The asymptotic properties of the estimators are given: (a) we provide the uniform weak convergence rates of the estimators of the trend function and periodic sequence; (b) we establish the asymptotic normality for the trend function estimator; and (c) we show that the period estimator is consistent. Plug-in type bandwidths are proposed for the trend estimator. A simulation study and an application to sea level data are employed in order to complement our theoretical results.