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Estimation for the autoregressive moving average model with a unit root

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We consider the model(UNFORMATTED TABLE OR EQUATION FOLLOWS) y[subscript]t&=[rho] y[subscript]t-1+z[subscript]t, z[subscript]t+[alpha][subscript]1z[subscript]t-1 +·s+[alpha][subscript]pz[subscript]t-p&= e[subscript]t+ [beta][subscript]1e[subscript]t-1+·s+[beta][subscript]qe[subscript]t-q, t=1,...,n, (TABLE/EQUATION ENDS)where e[subscript]t is an iid (0, [sigma][superscript]2) sequence and z[subscript]t is a stationary and invertible process. The parameter vector ([theta], [rho], [sigma]), where [theta]=([alpha][subscript]1, [alpha][subscript]2, ..., [alpha][subscript]p, [beta][subscript]1, [beta][subscript]2, ..., [beta][subscript]q)', can be estimated by least squares or by maximum likelihood, where the likelihood is constructed under the assumption that e[subscript]t is a sequence of normal (0, [sigma][superscript]2) random variables. We show that the least squares estimator and the maximum likelihood estimator of ([theta], [rho], [sigma]) are strongly consistent when [rho]=1. When the true value of [rho] is one, the limiting distributions of the two estimators of ([theta], [rho]) are the same. The limiting distribution of the estimator of [rho] is the same as the limiting distribution of the ordinary least squares estimator of [rho] in the first order autoregressive model. The limiting distribution of the estimator of [theta] is the same multivariate normal distribution as the limiting distribution of the estimator obtained with known [rho];Extensions for the model containing an intercept are investigated. We show that the estimators are weakly consistent and derive the limiting distribution of the estimators when the true value of [rho] is one. When [rho] = 1, the limiting distribution of the estimator of [rho] is the same as the limiting distribution of the ordinary least squares estimator of [rho] in the first order autoregressive model.