﻿ 单位根过程与分整过程间变化变点的Sieve Bootstrap检验 Sieve Bootstrap Test for Changes between Unit Root Process and Fractional Integrated Processes

Statistics and Application
Vol.07 No.02(2018), Article ID:24337,6 pages
10.12677/SA.2018.72014

Sieve Bootstrap Test for Changes between Unit Root Process and Fractional Integrated Processes

Mingcan He, Guolong Fu

School of Mathematics and Statistics, Qinghai Normal University, Xining Qinghai

Received: Mar. 15th, 2018; accepted: Apr. 2nd, 2018; published: Apr. 9th, 2018

ABSTRACT

This paper aimed to test change point from unit root process to fractional integrated process as well as fractional integrated process to unit process via a Dickey-Full ratio statistic. A Sieve Bootstrap method was proposed to determine the critical values. Simulations indicate that our proposed method can control the empirical size well both under the unit root and fractional integrated process null hypotheses, and gives satisfy empirical powers under two alternative hypotheses if the change point location does not too back. Furthermore, Dickey-Full ratio statistic has better performance when detecting those changes which from unit root process to fractional integrated process.

Keywords:Unit Root Process, Fractional Integrated Process, Changes Point, Sieve Bootstrap

Copyright © 2018 by authors and Hans Publishers Inc.

1. 引言

2. 模型与假设检验

${\left(1-L\right)}^{{d}_{0}}{X}_{t}={\epsilon }_{t},\text{ }\text{ }\text{ }\text{ }\text{ }t=1,2,\cdots ,n,$

${X}_{t}=\sum _{j=0}^{\infty }{w}_{j}\left({d}_{0}\right){\epsilon }_{t-j}.$

${X}_{t}~I\left(1\right)\left({d}_{0}=1\right),\text{\hspace{0.17em}}{X}_{t}~I\left({d}_{0}\right)\left(0<{d}_{0}<1,\text{ }\text{ }\text{ }且\text{ }\text{ }\text{ }{d}_{0}\ne 0\right)$

1) 序列 ${X}_{t}$$I\left(1\right)$$I\left({d}_{0}\right)$ 变化的持久性问题假设检验，即检验原假设

${H}_{0}^{1}:{X}_{t}~I\left(1\right),\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }t=1,2,\cdots ,n.$

${H}_{1}^{1}:{X}_{t}~\left\{\begin{array}{l}I\left(1\right),\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }t=1,\cdots ,{k}^{*},\\ I\left({d}_{0}\right),\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }t={k}^{\ast }+1,\cdots ,n.\end{array}$

2) 序列 ${X}_{t}$$I\left({d}_{0}\right)$$I\left(1\right)$ 变化的持久性问题假设检验，即检验原假设

${H}_{0}^{2}:{X}_{t}~I\left({d}_{0}\right),\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }t=1,2,\cdots ,n,$

${H}_{1}^{2}:{X}_{t}~\left\{\begin{array}{l}I\left({d}_{0}\right),\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }t=1,\cdots ,{k}^{*},\\ I\left(1\right),\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }t={k}^{\ast }+1,\cdots ,n.\end{array}$

${\Xi }_{n}\left(\tau \right)=|\frac{D{F}^{f}}{D{F}^{r}}|,$

3. Bootstrap方法

1) 估计 ${X}_{t}$ 的长记忆参数 ${d}_{0}$ ，并记估计值为 $\stackrel{^}{d}$

2) 对 ${X}_{t}$ 进行 $\stackrel{^}{d}$ 阶差分， ${\stackrel{^}{\epsilon }}_{t}={\left(1-L\right)}^{\stackrel{^}{d}}\left({X}_{t}-{X}_{t-1}\right),\text{\hspace{0.17em}}t=1,\cdots ,n$ 。其中 ${\left(1-L\right)}^{\stackrel{^}{d}}=\sum _{j=0}^{t}{\stackrel{^}{a}}_{j}{L}^{j},\text{\hspace{0.17em}}{a}_{0}=1$${a}_{j}={a}_{j-1}\left(j-\stackrel{^}{d}-1\right)/j,\text{\hspace{0.17em}}j\ge 1$

3) 对 ${\stackrel{^}{\epsilon }}_{1},\cdots ,{\stackrel{^}{\epsilon }}_{t}$ 进行重抽样，得到新的序列 ${\stackrel{^}{\epsilon }}_{ij}^{\ast },\cdots ,{\stackrel{^}{\epsilon }}_{nj}^{*}$

4) 生成Bootstrap样本 ${\stackrel{^}{Y}}_{tj}^{\ast }={\left(1-L\right)}^{-\stackrel{^}{d}}{\stackrel{^}{\epsilon }}_{tj}^{\ast }+{\stackrel{^}{z}}_{tj}$ ，其中 ${\left(1-L\right)}^{-\stackrel{^}{d}}=\sum _{s=0}^{t}{b}_{j}{L}^{j},\text{\hspace{0.17em}}{b}_{0}=1$

${\stackrel{^}{z}}_{tj}$ 是独立同分布的标准正态分布随机序列。

5) 计算统计量 ${\Xi }_{n}\left({\stackrel{^}{Y}}_{tj}^{\ast }\right)$ ，重复步骤3~5 B次，取 ${\Xi }_{n}\left({\stackrel{^}{Y}}_{tj}^{\ast }\right)$$1-\alpha$ 分位数作为检验统计量在显著水平a

4. 数值模拟

Table 1. Empirical sizes of Ξ n ( X ) (%)

Table 2. Power values of Ξ n ( X ) (%)

5. 总结

Sieve Bootstrap Test for Changes between Unit Root Process and Fractional Integrated Processes[J]. 统计学与应用, 2018, 07(02): 111-116. https://doi.org/10.12677/SA.2018.72014

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