Local linear m-estimators in null recurrent time series

In this paper, we study a nonlinear cointegration type model Y_k=m(X_k)+w_k , where \ Y_k\ and \ X_k\ are observed nonstationary processes and \ w_k\ is an unobserved stationary process. The process \ X_k\ is assumed to be a null-recurrent Markov chain. We apply a robust version of local linear regression smoothers to estimate m(\cdot) . Under mild conditions, the uniform weak consistency and asymptotic normality of the local linear M-estimators are established. Furthermore, a one-step iterated procedure is introduced to obtain the local linear M-estimator and the optimal bandwidth selection is discussed. Meanwhile, some numerical examples are given to show that the proposed theory and methods perform well in practice.