Tomoyoshi Yabu

Testing for Shifts in Trend With an Integrated or Stationary Noise Component

We consider testing for structural changes in the trend function of a time series without any prior knowledge of whether the noise component is stationary or integrated. Following Perron and Yabu (2009), we consider a quasi-feasible generalized least squares procedure that uses a super-efficient estimate of the sum of the autoregressive parameters αwhen α=1. This allows tests of basically the same size with stationary or integrated noise regardless of whether the break is known or unknown, provided that the Exp functional of Andrews and Ploberger (1994) is used in the latter case. To improve the finite-sample properties, we use the bias-corrected version of the estimate of αproposed by Roy and Fuller (2001). Our procedure has a power function close to that attainable if we knew the true value of αin many cases. We also discuss the extension to the case of multiple breaks.

Testing for Trend in the Presence of Autoregressive Error: A Comment

Roy, Falk and Fuller (2004) presented a procedure aimed at providing a test for the value of the slope of a trend function that has (nearly) controlled size in autoregressive models whether the noise component is stationary or has a unit root. In this note, we document errors in both their theoretical results and the simulations they reported. Once these are corrected for, their procedure delivers a test that has very liberal size in the case with a unit root so that the stated goal is not achieved. Interestingly, the mistakes in the code used to generate the simulated results (which is the basis for the evidence about the reliability of the method) are such that what they report is essentially equivalent to the size and power of the test proposed by Perron and Yabu (2009), which was shown to have the standard Normal distribution whether the noise is stationary or has a unit root.

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 I± when I±=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 I± 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.