The Variance Ratio Statistic At Large Horizons

We make three contributions to using the variance ratio statistic at large horizons. Allowing for general heteroskedasticity in the data, we obtain the asymptotic distribution of the statistic when the horizon k is increasing with the sample size n but at a slower rate so that k/n → 0. The test is shown to be consistent against a variety of relevant mean reverting alternatives when k/n → 0. This is in contrast to the case when k/n → δ > 0, where the statistic has been recently shown to be inconsistent against such alternatives. Second, we provide and justify a simple power transformation of the statistic that yields almost perfectly normally distributed statistics in finite samples, solving the well-known right skewness problem. Third, we provide a more powerful way of pooling information from different horizons to test for mean reverting alternatives. Monte Carlo simulations illustrate the theoretical improvements provided.The authors thank Bruce Hansen and the referees for useful suggestions and comments that greatly improved the paper. The first author's research was supported by NSF grant DMS-0306726.