On the Asymptotic Power of the Variance Ratio Test

The variance ratio 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 variance ratio test statistic when the differencing period k is increasing with the sample size n such that k/n → δ > 0. We show that the test is inconsistent against a variety of mean reverting alternatives, confirm the result in simulations, and then characterise the functional form of the asymptotic power in terms of δ and these alternatives