Testing for Flexible Nonlinear Trends with an Integrated or Stationary Noise Component

This paper proposes a new test for the presence of a nonlinear deterministic trend approximated by a Fourier expansion in a univariate time series for which there is no prior knowledge as to whether the noise component is stationary or contains an autoregressive unit root. Our approach builds on the work of Perron and Yabu (2009a) and is based on a Feasible Generalized Least Squares procedure that uses a super-efficient estimator of the sum of the autoregressive coefficients α when α=1. The resulting Wald test statistic asymptotically follows a chi-square limit distribution in both the I(0) and I(1) cases. To improve the finite sample properties of the test, we use a bias corrected version of the OLS estimator of α proposed by Roy and Fuller (2001). We show that our procedure is substantially more powerful than currently available alternatives. We illustrate the usefulness of our method via an application to modeling the trend of global and hemispheric temperatures.