Exact Variance Ratio Test with Overlapping Data

Abstract

Variance ratio test is popular in finance for testing whether stock prices follow random walk or not. However, this test is typically conducted based on its asymptotic distribution, making it difficult to draw exact inference. Under the multivariate elliptical distribution assumption but allowing for arbitrary variance-covariance matrix of returns, we provide analysis of the exact moments and distribution of the variance ratio test as well as delivering an efficient procedure to compute the exact p-value of the test statistic. Our results allow us to study the optimal length of multi-period return for detecting different alternatives to the random walk hypothesis of stock prices.