Statistica Sinica 19 (2009), 1683-1703

LOCAL LINEAR M-ESTIMATORS IN NULL

RECURRENT TIME SERIES

Zhengyan Lin$^{1}$, Degui Li$^{1, 2}$ and Jia Chen$^{1, 2}$

$^{1}$Zhejiang University and $^{2}$University of Adelaide

Abstract: 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.

Key words and phrases: Asymptotic normality, β-null recurrent Markov chain, cointegration model, consistency, local linear M-estimator.