Robust tests for a linear trend with an application to equity indices

In this paper we develop a testing procedure for the presence of a deterministic linear trend in a univariate time series which is robust to whether the series is I(0) or I(1) and requires no knowledge of the form of weak dependence present in the data. Our approach is motivated by the testing procedures of Vogelsang [1998, Econometrica, vol 66, p123–148] and Bunzel and Vogelsang [2005, Journal of Business and Economic Statistics, vol 23, p381–394], but utilises an auxiliary unit root test to switch between critical values in the exact I(1) and I(0) environments, rather than using this unit root test to scale the test statistic as is done in the aforementioned procedures. We show that our proposed tests have uniformly greater local asymptotic power than the tests of Vogelsang (1998) and Bunzel and Vogelsang (2005) when the error process is exact I(1), identical local asymptotic when the error process is I(0), and have better overall local asymptotic power when the error process is near I(1). Our proposed tests also display superior finite sample power to the tests of Vogelsang (1998) and Bunzel and Vogelsang (2005) and are competitive in finite samples with tests designed to be optimal in both the exact I(1) and I(0) environments. We apply our test procedures to a number of equity indices and find that these series appear to have a significant upward deterministic trend, yet are also highly persistent about this long run growth path.