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 = α + βt + u_t, where u_t follows an autoregressive process with largest root ρ and β is the parameter of interest. We contrast asymptotic results for the cases |ρ| < 1 and ρ = 1 and argue that the most useful asymptotic approximations obtain from modeling ρ 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 κ. 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 ρ and κ. The paper develops methods for constructing confidence intervals for β that account for uncertainty in ρ and κ. We use these results to estimate growth rates for real per-capita GDP in 128 countries.