Closed forms for asymptotic bias and variance in autoregressive models with unit roots

For a first-order autoregressive AR(1) model with zero initial value, xt = αxt−1 + εt, we provide closed-form analytical expressions for the asymptotic bias and variance of the maximum likelihood (ML) estimator α = ∑1n xtxt−1∑1n−1 xt2 when ¦α¦ = 1. For the bias, numerical accuracy of up to six significant digits is achieved for sample sizes n > 100.