| In recent years, the unit root test problem in time series theory is wildly studied by many researchers. As we know that if the regression model is unknown, we must first make unit root test in order to check the stationary of the model. Some results have been created and discussed when the errors are i.i.d. But in many cases, especially in some researches of economics and finances, the regression model are often with dependent residuals.The normal unit root test can not be used in this situation. So the bootstrap methods are studied to help us complete the unit root test.Prof. Efron introduce a new sample statistic method-the bootstrap: Let {Xk, 1≤k≤n} be the sample observations from population X,either independent or dependent. {m1, m2,…} is a integer sequence.we write that Xn = (X1, X2,…, Xn),for allmn, Xn,1*,Xn,2*,…,Xn,mn* denote the samples that we select from X1, X2,…, Xn. We call them the bootstrap samples with a sample size of mn. Next we use the bootstrap method in the unit root test when the residuals are dependent.For a real AR(1) model which is a random walk:Xi= Xi-1+εi, X0= 0, i∈NBut in some cases we do not know the true model, so we may choose different regression model under different asumptions.For the regression model without a constant:Xi =θXi-1+εi, X0= 0, i∈N,θ∈R.we make the unit root test with null hypothesisH0 :θ= 1Horvath and Kokoszka(2003)proved that the null distribution of the unit root test statistic based on the least-square estimator can be approximated by using residual bootstrap whenεi are in the domain of attraction of a strictly a-stable law.Analogously, Zhang LX, Yang XR (2006)show the bootstrap distribution when the residualsεi is a stationary sequence of mean zero and finite variance.in this paper, we mainly consider the bootstrap distribution when the regression model include a constant,i.e.Xi =α+θXi-1 +εi, X0= 0, i∈N,α∈R,θ∈R. Respectively,we make the unit root test under null hypothesisH0 :α= 0且θ= 1.First we can get the LSE ofθandαwhereNow we describe the bootstrap procedure as the following steps:1. Calculate the residuals:ε1* = X1- (?)n,ε2* = X2-(?)nX1-(?)n,…,εn* =Xn-(?)nXn-1-(?)n where (?)n and (?)n are the estimators defined above.2.For a fixed m < n, select a bootstrap sample (?)1, (?)1,…, (?)m fromε1*,ε2*,…,εn*3.Construct the bootstrap process: (?)0 = 0, (?), 1≤i≤m, andcalculate the estimatorswhere4.get the distribution of m((?)m -(?)n) and m1/2((?)m - (?)n)At the end of this paper we conclude our main result as follows:where...
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