A Joint Variance Ratio Test based on the Wild Bootstrap

The variance ratio (VR) test has been widely used as a means of testing for the martingale hypothesis in financial time series. The conventional VR tests are asymptotic tests, which can exhibit deficient small sample properties. In this paper, a joint VR test based on the wild bootstrap is proposed. It is a small sample test which does not rely on asymptotic approximations. An extensive Monte Carlo experiment is conducted to evaluate and compare the small sample properties of alternative joint VR tests including the wild bootstrap version. It is found that the wild bootstrap test is the only test which shows desirable size properties under a wide range of data generation processes. It also shows excellent power properties for a variety of non-martingale alternatives including long memory time series. As an application, the martingale hypothesis in major exchange rates is tested using the wild bootstrap test.