Distribution of the Estimators for Autoregressive Time Series With a Unit Root
Let n observations Y<sub>1</sub>, Y<sub>2</sub>, ..., Y<sub>n</sub> be generated by the model Y<sub>t</sub> = ρ Y<sub>t - 1</sub> + e<sub>t</sub>, where Y<sub>0</sub> is a fixed constant and e<sub>t</sub><sub>t = 1</sub><sup>n</sup> is a sequence of independent normal random variables with mean 0 and variance σ<sup>2</sup>. Properties of the regression estimator of ρ are obtained under the assumption that ρ = ± 1. Representations for the limit distributions of the estimator of ρ and of the regression t test are derived. The estimator of ρ and the regression t test furnish methods of testing the hypothesis that ρ = 1.