Abstract

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.

Highlights

  • Since the random walk is a martingale sequence, the best predictor of the term becomes the value of this term

  • We review the extension of the FCLT results to the cases that the innovations are locally stationary process

  • We investigate the asymptotic properties of least squares estimators for unit root, near unit root, and I d processes with locally stationary process innovations

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Summary

Introduction

Since the random walk is a martingale sequence, the best predictor of the term becomes the value of this term. We review the fundamental asymptotic results for unit root processes. Random walk is included in unit root I 1 processes which is a class of nonstationary processes. L RT −→ L W as T −→ ∞, 1.3 where L · denotes the distribution law of the corresponding random elements This result is the so-called functional central limit theorem FCLT see Billingsley 1. We can derive the limiting distribution of LSE of unit root process from the FCLT result. For more detailed understanding about unit root process with i.i.d. or stationary innovation, refer to, for example, Billingsley 1 and Tanaka 3. We review the extension of the FCLT results to the cases that the innovations are locally stationary process.

The Property of Least Squares Estimator
Least Squares Estimator for Unit Root Process
Least Squares Estimator for Near Unit Root Process
Testing for Unit Root
Ochi Estimator
Tθ Tj 1 j T
FCLT for Locally Stationary Processes
Unit Root Process with Locally Stationary Disturbance
Near Unit Root Process with Locally Stationary Disturbance
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