Properties of the Power Envelope for Tests Against Both Stationary and Explosive Alternatives: The Effect of Trends

This article details a precise analytic effect 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 difference 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 effect 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 specific rate effect on the power envelope, however other deterministic variables will have some effect. The methods of the article lead to a simple direct measure of this effect which is then informative about power, in practice.