FINITE SAMPLE DISTRIBUTIONS OF THE OLS ESTIMATOR IN A FIRST ORDER AUTOREGRESSIVE MODEL.

Abstract

It is well-known that the least-square estimator φˆ in the firstorder autoregressive model for the unit root case is biased in finite samples and its exact distribution remains hitherto unknown. We propose a new methodology to derive closed-form expressions to analyse the finite-sample distributional properties of φˆ corresponding to the unit root case where the initial value y0 is a constant with probability one. Using this framework, we can evaluate higher-order moments, critical values as well as the cumulative distribution function corresponding to various sample sizes and initial values through the use of numerical integration of closed-form integral expressions, which are smooth and continuous. In addition, a novel asymptotic closed-form expression for the first moment is derived. Our results are in close agreement with the values obtained in the survey of the existing literature.