Saddlepoint Approximations for the Unit Root Power Envelope

The unit root power envelope is used both as a benchmark and as a mechanism for generating feasible unit root tests, via quasi-differencing. This paper derives an explicit representation for the envelope, via direct saddlepoint expansions for the inversion formulae for both the null and alternative distributions of the set of point optimal tests. It is shown to be both more accurate and computationally efficient than the current partial sum based approximations to limiting representations in terms of stochastic integrals. Accuracy is demonstrated through a sequence of experiments and efficiency via application to find the efficient detrending parameter in models with broken trends.