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.