Estimation for autoregressive time series with a root near 1

Estimators for the parameters of autoregressive time series are compared, e mphasizing processes with a unit root or a root close to 1. The approximate bias of the sum of the autoregressive coefficients is expressed as a funct ion of the test for a unit root. This expression is used to construct an es timator that is nearly unbiased for the parameter of the first-order scalar process. The estimator for the first-order process has a mean squared erro r that is about 40% of that of ordinary least squares for the process with a unit root and a constant mean, and the mean squared error is smaller than that of ordinary least squares for about half of the parameter space. The maximum loss of efficiency is 6n(-1) in the remainder of the parameter spac e. The estimation procedure is extended to higher-order processes by modify ing the estimator of the sum of the autoregressive coefficients. Limiting r esults are derived for the autoregressive process with a mean that is a lin ear trend.