Testing for the Martingale Hypothesis

This paper proposes general speci...cation tests for the martingale hypothesis. They can be used to test the null hypothesis that a given time series is a martingale process, against the alternative hypothesis that it is a stationary ergodic nonmartingale process. We consider tests of two di¤erent types: one is a generalized Kolmogorov-Smirnov test and the other is a Cramer-von Mises type test. The tests introduced in the paper are simple to compute, neither depend upon any smoothing parameter nor require any resampling procedure to simulate the null distributions. The null distributions of the test statistics are given by the functionals of a continuous martingale process, which are free of any nuisance parameter. We show that the tests are consistent against stationary ergodic non-martingale alternatives, and further investigate the ...nite sample properties of the test statistics through simulation. Our tests are found to be rather powerful in moderate size samples against a wide variety of non-martingales including exponential autoregressive, threshold autoregressive, markov switching, and chaotic processes. First Draft: June 10, 1999 This version: December 1, 1999