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The test, proposed by Moore and Wallis [23], of the existence of correlation between the movements in two time series (eliminating at least the primary effects of trends) was justified by them in "the special case of randomly arranged [signs of the] first differences" in both series. Moore and Wallis also noted that the signs of the first differences in a time series will have a negative serial correlation if the time series is a purely random process (or if it is a particular kind of generalization of a purely random process). It is therefore necessary to modify the Moore-Wallis test when applying it to such time series in order to take into account this serial correlation. In the present article, a simple modified form of the Moore-Wallis test is suggested, which can be applied to such time series. Somewhat similar modifications are also suggested for testing the existence of correlation between some more general kinds of time series. The importance of taking into account the effects of trends in the time series and the effects of the serial correlations (between the signs of the first differences) is illustrated by means of two numerical examples. Finally, the study of the movements in three or more time series is briefly discussed.

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