This paper details a precise analytic e¤ect that inclusion of a linear trend has on the power of Neyman-Pearson point optimal unit root tests and thence the power envelope. Both stationary and explosive alternatives are considered. The envelope can be characterized by probabilities for two, related, sums of chi-square random variables. A stochastic expansion, in powers of the local-to-unity parameter, of the di¤erence between these loses its leading term when a linear trend is included. This implies that the power envelope converges to size at a faster rate, which can then be exploited to prove that the power envelope must necessarily be lower. This e¤ect is shown to be, analytically, greater asymptotically than in small samples and numerically far greater for explosive than for stationary alternatives. Only a linear trend has a speci…c rate e¤ect on the power envelope, however other deterministic variables will have some e¤ect. The methods of the paper lead to a simple direct measure of this e¤ect which is then informative about power, in practice.
Marsh, P. (2020). Properties of the Power Envelope for Tests Against Both Stationary and Explosive Alternatives: The Effect of Trends. Journal of Time Series Analysis, 41(1), 146-153. https://doi.org/10.1111/jtsa.12458