Exact non-parametric tests for a random walk with unknown drift under conditional heteroscedasticity

This paper proposes a class of linear signed rank statistics to test for a random walk with unknown drift in the presence of arbitrary forms of conditional heteroscedasticity. The class considered includes analogues of the well-known sign and Wilcoxon test statistics. The exactness of the proposed tests rests only on the assumption that the errors are symmetrically distributed. No other assumptions, such as normality or even the existence of moments, are required. These tests extend the non-parametric approach to testing for a random walk proposed in Campbell and Dufour (International Economic Review 38 (1997) 151). Simulations confirm the reliability of the new tests, and their power can be considerably superior to the bounds tests of Campbell and Dufour, especially for alternatives close to the null. The inference methods developed are illustrated by a test of the random walk hypothesis in exchange rates for five major currencies against the U.S. dollar.