The size and power of the variance ratio test in finite samples

Abstract

We examine the finite-sample properties of the variance ratio test of the random walk hypothesis via Monte Carlo simulations under two null and three alternative hypotheses. These results are compared to the performance of the Dickey–Fuller t and the Box–Pierce Q statistics. Under the null hypothesis of a random walk with independent and identically distributed Gaussian increments, the empirical size of all three tests are comparable. Under a heteroscedastic random walk null, the variance ratio test is more reliable than either the Dickey–Fuller or Box–Pierce tests. We compute the power of these three tests against three alternatives of recent empirical interest: a stationary AR(1), the sum of this AR(1) and a random walk, and an integrated AR(1). By choosing the sampling frequency appropriately, the variance ratio test is shown to be as powerful as the Dickey–Fuller and Box–Pierce tests against the stationary alternative and is more powerful than either of the two tests against the two unit root alternatives.