Table of Contents
Advances in Decision Sciences
Volume 2012, Article ID 893497, 16 pages
Research Article

Least Squares Estimators for Unit Root Processes with Locally Stationary Disturbance

Faculty of Science, Niigata University, 8050, Ikarashi 2-no-cho, Nishi-ku, Niigata 950-2181, Japan

Received 4 November 2011; Accepted 26 December 2011

Academic Editor: Hiroshi Shiraishi

Copyright © 2012 Junichi Hirukawa and Mako Sadakata. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The random walk is used as a model expressing equitableness and the effectiveness of various finance phenomena. Random walk is included in unit root process which is a class of nonstationary processes. Due to its nonstationarity, the least squares estimator (LSE) of random walk does not satisfy asymptotic normality. However, it is well known that the sequence of partial sum processes of random walk weakly converges to standard Brownian motion. This result is so-called functional central limit theorem (FCLT). We can derive the limiting distribution of LSE of unit root process from the FCLT result. The FCLT result has been extended to unit root process with locally stationary process (LSP) innovation. This model includes different two types of nonstationarity. Since the LSP innovation has time-varying spectral structure, it is suitable for describing the empirical financial time series data. Here we will derive the limiting distributions of LSE of unit root, near unit root and general integrated processes with LSP innovation. Testing problem between unit root and near unit root will be also discussed. Furthermore, we will suggest two kind of extensions for LSE, which include various famous estimators as special cases.