An invariant sign test for random walks based on recursive median adjustment

We propose a new invariant sign test for random walks against general stationary processes and develop a theory for the test. In addition to the exact binomial null distribution of the test, we establish various important properties of the test: the consistency against a wide class of possibly nonlinear stationary autoregressive conditionally heteroscedastic processes and/or heavy-tailed errors; a local asymptotic power advantage over the classical Dickey–Fuller test; and invariance to monotone data transformations, to conditional heteroscedasticity and to heavy-tailed errors. Using the sign test, we also investigate various interrelated issues such as M-estimator, exact confidence interval, sign test for serial correlation, robust inference for a cointegration model, and discuss possible extensions to models with autocorrelated errors. Monte-Carlo experiments verify that the sign test has not only very stable sizes but also locally better powers than the parametric Dickey–Fuller test and the nonparametric tests of Granger and Hallman (1991. Journal of Time Series Analysis 12, 207–224) and Burridge and Guerre (1996. Econometric Theory 12, 705–719) for heteroscedastic and/or heavy tailed errors.