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I'm working on the multiple price series data to look for the long run relationships. DF-GLS tests for unit root are rejected although the series show some trends and seem stationary after differenced. Yet Engle-Granger tests suggest that the series are indeed cointegrated.

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You can run the Engle-Granger procedure only if the series are integrated. Meanwhile, if you have stationary series and (ignoring the absence of unit roots) run the Engle-Granger procedure nevertheless, you will find that the residual from the relevant regression is stationary and will be tempted to conclude that cointegration exists. But that is not warranted if the series are not integrated to begin with; without that the Engle-Granger procedure does not make sense.

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