We propose several multivariate variance ratio statistics. We derive the asymptotic distribution of the statistics and scalar functions thereof under the null hypothesis that returns are unpredictable after a constant mean adjustment (i.e., under the weak form Efficient Market Hypothesis). We do not impose the no leverage assumption of Lo and MacKinlay (1988) but our asymptotic standard errors are relatively simple and in particular do not require the selection of a bandwidth parameter. We extend the framework to allow for a time varying risk premium through common systematic factors. We show the limiting behaviour of the statistic under a multivariate fads model and under a moderately explosive bubble process: these alternative hypotheses give opposite predictions with regards to the long run value of the statistics. We apply the methodology to five weekly size-sorted CRSP portfolio returns from 1962 to 2013 in three subperiods. period, for small and medium cap stocks. The main findings are not substantially affected by allowing for a common factor time varying risk premium
