Estimating deterministic trends in the presence of serially correlated errors

This paper studies the problems of estimation and inference in the linear trend model y(t) = alpha + beta t + u(t), where u(t) follows an autoregressive process with largest root rho and beta is the parameter of interest. We contrast asymptotic results for the cases \rho\ < 1 and rho = 1 and argue that the most useful asymptotic approximations obtain from modeling rho as local to unity. Asymptotic distributions are derived for the OLS, first-difference, infeasible GLS, and three feasible GLS estimators. These distributions depend on the local-to-unity parameter and a parameter that governs the variance of the initial error term kappa. The feasible Cochrane-Orcutt estimator has poor properties, and the feasible Prais-Winsten estimator is the preferred estimator unless the researcher has sharp a priori knowledge about rho and kappa. The paper develops methods for constructing confidence intervals for beta that account for uncertainty in rho and kappa. We use these results to estimate growth rates for real per-capita GDP in 128 countries.