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When doing cointegration tests should I use the adjusted close price or just close price for the time series? The dividend of each stock is on different dates and can cause jumps in the data.

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Such tests should always be done using adjusted prices. In fact, ideally, you should reconstruct your own price series using the total returns series. To see this, suppose you have a 10:1 split rather than a relatively small cash dividend. Then it is clear that the cointegration relationship can only hold with respect to the adjusted series.

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