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

Abstract

Let n observations Y ₁, Y ₂, ···, Y _n be generated by the model Y _t = pY _t−1 + e _t , where Y ₀ 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 p are obtained under the assumption that p = ±1. Representations for the limit distributions of the estimator of p and of the regression t test are derived. The estimator of p and the regression t test furnish methods of testing the hypothesis that p = 1.