Adjusted estimates and Wald statistics for the AR(1) model with constant

Adjusted profile likelihood of Cox and Reid (1993, Journal of the Royal Statistical Society B 55, 467–471) for the autoregressive coefficient of the AR(1) model with constant yields an estimator which can be more accurate than the maximum-likelihood estimator in small samples whether the process is stationary or a random walk, and in the latter case also asymptotically. A related estimator is asymptotically distributed compactly and symmetrically around zero under a random walk but is not consistent in general. The adjusted Wald statistic can follow the asymptotic standard normal distribution much better than the non-adjusted under stationarity or a drifted random walk.