The Cowles–Jones test with unspecified upward market probability

Abstract

<abstract><p>The Cowles-Jones test for sign dependence is one of the earliest tests of the random walk hypothesis, which stands at the beginning of modern empirical finance. The test is still discussed in popular textbooks and used in research articles. However, the Cowles-Jones test statistic considered in the literature requires that the upward probability of the market or asset under consideration be specified under the null hypothesis, which is only very rarely possible. If the upward probability is estimated in advance, the resulting test is undersized (even asymptotically). This note considers a corrected Cowles-Jones test statistic which does not require the upward probability to be specified under the null. It turns out that the asymptotic variance is greatly simplified as compared to the uncorrected test. The corrected test is illustrated with an application to daily returns of the Dow Jones Industrial Average index and monthly returns of the MSCI Emerging Markets index. It is shown that the corrected and uncorrected tests can lead to opposite conclusions.</p></abstract>