A Permutation Test for Unit Root in an Autoregressive Model

Jiexiang Li,Sa-Aat Niwitpong,Lanh Tran

doi:10.4236/am.2013.412221

Jiexiang Li, Sa-Aat Niwitpong + Show 1 more

Open Access

https://doi.org/10.4236/am.2013.412221

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Abstract

A permutation test (based on a finite random sample of permutations) for unit root in an autoregressive process is considered. The test can easily be carried out in practice and the proposed permutation test is neither limited to large sample sizes nor normal white noises. Simulations show that the power of the permutation test is reasonable when sample sizes are small or when the white noises have a heavy tailed distribution. The test is shown to be consistent.

Highlights

Let Y1,Y2,Yn 1 be n 1 observations of the real valued autoregressive modelH0 : a 1 versus H A : 0 < a < 1, are often referred to as tests for unit root
When sample sizes are small and white noises are from distributions with heavy tails, to test for the presence of unit root, we can use the permutation test proposed in the paper
We will show that the permutation test based on random sampling R permutations is consistent, that is, the probability of correctly rejecting the null hypothesis H0 tends to 1 as the sample size goes to infinity when 0 < a < 1

Summary

Introduction

H0 : a 1 versus H A : 0 < a < 1, are often referred to as tests for unit root. The hypothesis that a 1 is of interest in applications because it corresponds to the hypothesis that it is appropriate to transform the times series by difference. When sample sizes are small and white noises are from distributions with heavy tails, to test for the presence of unit root, we can use the permutation test proposed in the paper. In Economics, it is important to characterize the velocity, or stock price as a random walk Another way of phrasing H0 is that whatever determinants of velocity and their individual stochastic structure may be, their combined effect is such that successive changes in velocity are essentially independent. Statistic Tn under permutations when H0 holds, we propose a permutation test for unit root using Tn as our pivot test statistic This test is easy to perform with a computer and the test makes little assumptions on the probability distribution of the white noise and it works for small samples.

Steps Used in Our Permutation Test

Consistency of R

E X1 X 2 X j X j 1

E X1 X 2

Findings

An Example