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The main variable I'm investigating is a dummy variable that is 1 after a certain date and 0 before, denoting when a certain law passed. When I include date, date is significant and the dummy is not, indicating the change in response after the dummy date can explained by the existing time trend. But when I take out the date variable, the dummy variable is significant.

There is no obvious reason that date/time should be having an effect.

Should I keep date in the model?

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You give very little information regarding the time series (does it have seasonality for example?). In my opinion, the correct way to assess the impact of the regulation change in your series is to include a dummy variable and remove the date. You can also test for structural breaks in the time series. I believe the CUMSUM test is the most widely used in this case.

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