Wald-Type Tests for Detecting Breaks in the Trend Function of a Dynamic Time Series

In this paper, test statistics for detecting a break at an unknown date in the trend function of a dynamic univariate time series are proposed. The tests are based on the mean and exponential statistics of Andrews and Ploberger (1994, <italic>Econometrica</italic> 62, 1383–1414) and the supremum statistic of Andrews (1993, <italic>Econometrica</italic> 61, 821–856). Their results are extended to allow trending and unit root regressors. Asymptotic results are derived for both <italic>I</italic>(0) and <italic>I</italic>(1) errors. When the errors are highly persistent and it is not known which asymptotic theory (<italic>I</italic>(0) or <italic>I</italic>(1)) provides a better approximation, a conservative approach based on nearly integrated asymptotics is provided. Power of the mean statistic is shown to be nonmonotonic with respect to the break magnitude and is dominated by the exponential and supremum statistics. Versions of the tests applicable to first differences of the data are also proposed. The tests are applied to some macroeconomic time series, and the null hypothesis of a stable trend function is rejected in many cases.