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.

Since the random walk is a martingale sequence, the best predictor of the next term becomes the value of this term. In this sense, the random walk is used as a model expressing equitableness and the effectiveness of various finance phenomena in economics. Furthermore, because the random walk is a unit root process, taking the difference of the random walk, we can recover the independent sequence. However, the information of the original sequence will be lost by taking the difference when it does not include a unit root. Therefore, the testing of the existence of unit root in the original sequence becomes important.

In this section, we review the fundamental asymptotic results for unit root processes. Let

The FCLT result can be extended to the unit root process where the innovation is general linear process. We consider a sequence

The asymptotic property of LSE for stationary autoregressive models has been well established (see, e.g., Hannan [

In the above case, the

This paper is organized as follows. In the appendix, we review the extension of the FCLT results to the cases that the innovations are locally stationary process. Namely, we explain the FCLT for unit root, near unit root, and general integrated processes with LSP innovations. In Section

In this section, we investigate the asymptotic properties of least squares estimators for unit root, near unit root, and

Here, we consider the following statistics:

We next consider the least squares estimator

Furthermore, we consider the least squares estimator

In the analysis of empirical financial data, the existence of the unit root is an important problem. However, as we see in the previous section, the asymptotic results between unit root and near unit root processes are quite different (the drift term appeared in the limiting process of near unit root). Therefore, we consider the following testing problem against the local alternative hypothesis:

In this section, we consider the extensions of LSE

Ochi [

Define for

Next, we suggest another class of estimators which are also the extensions of LSE. Define for

Define for

In this appendix, we review the extensions of functional central limit theorem to the cases that innovations are locally stationary processes, which are used for the main results of this paper.

Hirukawa and Sadakata [

First, we introduce locally stationary process innovation. Let

We consider a sequence

In this section, we consider the following near unit root process

Let

For the general integer

The authors would like to thank the referees for their many insightful comments, which improved the original version of this paper. The authors would also like to thank Professor Masanobu Taniguchi who is the lead guest editor of this special issue for his efforts and celebrate his sixtieth birthday.