Distribution of the Estimators for Autoregressive Time Series With a Unit Root

Let n observations Y₁, Y₂, ..., Y_n be generated by the model Y_t = ρ Y_{t - 1} + e_t, where Y₀ is a fixed constant and e_t_{t = 1}ⁿ is a sequence of independent normal random variables with mean 0 and variance σ². 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.