A Class of Simple Semiparametrically Efficient Rank-Based Unit Root Tests

Abstract

We propose a class of simple rank-based tests for the null hypothesis of a unit root. This class is indexed by the choice of a reference density g, which needs not coincide with the unknown actual innovation density f. The validity of these tests, in terms of exact finite sample size, is guaranteed by distribution-freeness, irrespective of the value of the drift and the actual underlying f. When based on a Gaussian reference density g, our tests (of the van der Waerden form) perform uniformly better, in terms of asymptotic relative effciency, than the Dickey and Fuller test --except under Gaussian f, where they are doing equally well. Under Student t3 density f, the effciency gain is as high as 110%, meaning that Dickey-Fuller requires over twice as many observations as we do in order to achieve comparable performance. This gain is even larger in case the underlying f has fatter tails; under Cauchy f, where Dickey and Fuller is no longer valid, it can be considered infinite. The test associated with reference density g is semiparametrically e±cient when f happens to coincide with g, in the ubiquitous case that the model contains a non-zero drift. Finally, with an estimated density f(n) substituted for the reference density g, our tests achieve uniform (with respect to f) semiparametric efficiency.