 Applied Mathematics, 2013, 4, 1629-1634 Published Online December 2013 (http://www.scirp.org/journal/am) http://dx.doi.org/10.4236/am.2013.412221 Open Access AM A Permutation Test for Unit Root in an Autoregressive Model Jiexiang Li1, Lanh Tran2, Sa-aat Niwitpong3 1Department of Mathematics, College of Charleston, Charleston, USA 2Department of Statistics, Indiana University, Bloomington, USA 3Department of Applied Statistics, King Mongkut’s University of Technology, Bangkok, Thailand Email: lij@cofc.edu, tran@indiana.edu, snw@kmutnb.ac.th Received September 23, 2013; revised October 23, 2013; accepted October 30, 2013 Copyright © 2013 Jiexiang Li et al. 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. ABSTRACT A permutation test (based on a finite random sample of permutations) for unit root in an autoregressive process is con- sidered. 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. Keywords: Permutation Test; Autoregressive; Nonstationary 1. Introduction Let 12 1 be observations of the real valued autoregressive model ,,,nYY Y1n1,ttYaY et for and t is a sequence of independent identically distributed random variables with mean zero and variance 0< 1ae2 and . Tests of 10Y0:1versus: 0<< 1,AHaH a are often referred to as tests for unit root. The hypothesis that is of interest in applications because it corresponds to the hypothesis that it is appropriate to transform the times series by difference. In  and , the authors derived the limit distribution of the statistic with 1a1ˆna11121122ˆnnttitaY YYt under the unit root assumption . However,  and  are limited to large sample sizes or normal white noises. When sample sizes are small and white noises are from distributions with heavy tails, to test for the pres- ence of unit root, we can use the permutation test pro- posed in the paper. 1aUnder 0H, t is not stationary and the variance of is tYtY2. When 0H is true, is sometimes called a tYrandom walk and 1tt tXYY, are independent identically distributed r.v.’s. In Economics, it is important to characterize the velocity, or stock price as a random walk. Another way of phrasing 01, 2,,tnH 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 inde- pendent. This would imply that of the past history avail- able at any given date only the current observation is relevant for prediction. Testing for unit root is equivalent to testing for serial independence in sequence 1n,,XX. For some literature on tests for serial dependence, see  and  and the references therein. We define 111,nniiiTXX and for each random permutation of the vector 1,,nXX, say, 1,,nXX, denote 111.nniiiTXX If 0H holds then  12,, ,,nnXXee1. The distribution of a random vector of i.i.d. random variables is invariant under any permutation of its coordinates. Thus n and nT have the same distribution if 0TH is true. Based on the invariance of the distribution of the J. X. LI ET AL. 1630 statistic n under permutations when 0TH holds, we propose a permutation test for unit root using n 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 also it works for small samples. Construction of this test will be presented in Section 2. The consistency of our test is shown in Section 3. A simulation study of our test is provided in Section 4. Time series of velocity of money observed between 1869 and 1960 is investigated in Sec- tion 5. T2. Steps Used in Our Permutation Test We assume that the white noise te is a sequence of independent identically distributed random variables with mean zero and variance 2. In addition, 4