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I have a multiple regression where all my coefficients are significant. When I add new variables my initial variables become insignificant. Furthermore, my new variables (that in a simple regression are significant) turn out to be also insignificant.

My questions are:

  1. What are the possible reasons for this change of significance?
  2. What are the methods and possible solutions to solve this issue?
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    $\begingroup$ almost certainly the additional predictors are strongly related to some linear combination of your original predictors. This issue is discussed in many answers on site (try a search on multicollinear or multicollinearity for example) $\endgroup$ – Glen_b -Reinstate Monica Jan 14 '16 at 2:14
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    $\begingroup$ If it is about p=0.04 becoming 0.06, then the solution is to not attach too much importance to that. 0.04 does not prove a variable is important and 0.06 does not make it unimportant. $\endgroup$ – Björn Jan 14 '16 at 7:08
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As @Glen_b already said it is certainly that the variables are strongly related, so you have multicollinearity. You can make a correlation matrix for the variables, to check if the variables are indeed strongly related. Based on the correlation values, you could add interaction-terms to your model between variables that are highly correlated, as these might be significant.

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