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I have just made a regression with 2 explanatory variables and one dependent variable. One of the variables have a p-value of 0.0303 which I consider significant and the other have a p-value of 0.708. Since the variable with the biggest p-value does not impact my Y. Should I actually include both of the variables when writing the equation or only the one with the smallest p-value?

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If your question is about presenting the results - then you should write the whole equation, i.e., with both the significant and insignificant variables. That is your estimated model. If you omit the insignificant variable, you have to run the regression once again as the coefficient of the significant variable may change.

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