The Variance Ratio Test: An Analysis Of Size And Power Based On A Continuous-Time Asymptotic Framework

We consider the statistical properties of the variance ratio statistic in the context of testing for market efficiency defined by uncorrelated returns. The statistic is then the ratio of the variance of K-period returns to the variance of one-period returns scaled by K. We use a continuous-time asymptotic framework whereby we let the sample size increase to infinity keeping the span of the data fixed. We also let the aggregation parameter K increase such that K/T → κ as T, the sample size, increases to infinity. We consider the limit of the statistic under the null hypothesis and under three alternative hypotheses that have been popular in the finance literature. Our analysis permits us to address size and power issues with respect to κ and the sampling interval used. Our theoretical and simulation results show that power is initially increasing as κ increases but then decreases with further increases in κ. This shows that for any given alternative there exists a value of K relative to T that will maximize power. We thus investigate the properties of a test that is the maximal value of the variance ratio over a range of possible values for K. The importance of the trimming to define this range is highlighted.This paper is drawn from chapter 2 of Vodounou's Ph.D. dissertation at the Université de Montréal (Vodounou, 1997). We thank two referees for useful comments and Tomoyoshi Yabu and Xiaokang Zhu for research assistance. Perron acknowledges financial support from the Social Sciences and Humanities Research Council of Canada (SSHRC), the Natural Sciences and Engineering Council of Canada (NSERC), the Fonds pour la Formation de Chercheurs et l'Aide à la Recherche du Québec (FCAR), and the National Science Foundation (grant SES-0078492).