On the asymptotic power of the variance ratio test

The variance-ratio (VR) test statistic, which is based on k-period differences of the data, is commonly used in empirical finance and economics to test the random walk hypothesis. We obtain the asymptotic power function of the VR test statistic when the differencing period k is increasing with the sample size n such that k/n --> delta > 0. We show that the test is inconsistent against a variety of meanreverting alternatives, confirm the result in simulations, and then characterize the functional form of the asymptotic power in terms of delta and these alternatives.