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I have a set of predictors in a linear regression, as well as three control variables. The issue here is that one of my variables of interest is only statistically significant if the control variables are included in the final model. However, the control variables themselves are not statistically significant.

Here is how the multicollinearity of all my variables look like (including control variables):

 > vif(lm(return ~ EQ + EFF + SIZE + MOM + MSCR + UMP, data = as.data.frame(port.df)))
       EQ      EFF     SIZE      MOM     MSCR      UMP 
 3.687171 3.481672 2.781901 1.064312 1.438596 1.003408

 > vif(lm(return ~ EQ + MOM + MSCR, data = as.data.frame(port.df)))
       EQ      MOM     MSCR 
 1.359992 1.048142 1.412658 

My variables of interest are EQ, MOM and MSCR, and the control variables are EFF, SIZE and UMP. EQ is only significant if the three control var are included, and becomes insignificant when they are not:

  • Here are the coefficients (1rst row) and t-stats (2nd row) when control variables are included (notice that EQ is statistically significant)

           intercept           EQ          EFF        SIZE         MOM       MSCR          UMP
    [1,] 0.005206246 -0.006310531 0.0001229055 0.004125551 0.007738259 0.00473377 5.838596e-06
    [2,] 1.866628909 -1.746583234 0.0388823612 1.178460997 2.145062820 2.08131100 1.994863e-01
    
  • Now, here is the result of the regression when the control variables are excluded (notice that EQ is NOT statistically significant anymore)

           intercept           EQ         MOM       MSCR
    [1,] 0.007313402 -0.002111833 0.007128606 0.00668364
    [2,] 2.652662996 -0.595391117 2.036985378 2.80177366
    

The problem is that when I include my control variables, all my variables of interest are significant, but my control variables are not.

Which variables should I include in my final model? How should I structure my final model then, given the fact that the model will be used for forecasting?

Thank you,

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    $\begingroup$ Why are some variables "of interest" & some "control"? That is, if the point of the model's to forecast, why aren't they on an equal footing? Will some not be available for forecasting purposes? $\endgroup$ Sep 6, 2013 at 14:31
  • $\begingroup$ They will all be available for forecasting purposes. Should I then consider them all "of interest"? If that's the case, how would I deal with the fact that 3 of them are insignificant, but their presence affects the significance of another variable? $\endgroup$
    – Mayou
    Sep 6, 2013 at 14:33
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    $\begingroup$ While there may be a distinction between control and other variables in your mind, it is vital to realise that neither the statistics nor the software pays any attention to that distinction. $\endgroup$
    – Nick Cox
    Sep 6, 2013 at 14:39
  • $\begingroup$ True that. The question remains that even assuming all of these variables are "of interest", how should I structure my final model given the statistical significance shown above (that depends on which variables are included)? $\endgroup$
    – Mayou
    Sep 6, 2013 at 14:43
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    $\begingroup$ Just read up on LASSO again & forget about "significance". This is like a doctor asking how many leeches to apply after a course of antibiotics. $\endgroup$ Sep 6, 2013 at 15:14

2 Answers 2

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One reason to include control variables is precisely because they can affect other variables. In this case, the statistical significance of the control variable is completely irrelevant.

However, you may run into journal editors who disagree.

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  • $\begingroup$ Thank you for your comment! So how would you interpret the presence of the control variables? Also, does it make sense to use the model with insignificant control variables to forecast the response variable return? $\endgroup$
    – Mayou
    Sep 6, 2013 at 14:30
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    $\begingroup$ Interpreting the model is up to you. :-). I don't know what any of the variables actually are, and you are the one who knows the substantive area. The control variables, statistically, are there because you want to, well.... control for them! $\endgroup$
    – Peter Flom
    Sep 6, 2013 at 14:43
  • $\begingroup$ @PeterFlom Thank you for your answer. It is a great help! Could you also advise what to do if one runs into such journal editors? $\endgroup$
    – PSY
    Dec 10, 2021 at 20:22
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Just a short comment: your p-values should reflect the number of models you are "trying out". In some ways your approach of trying models with and without subsets of variables is one aspect of p-hacking. Your research question alone (not the data) should determine what is a control variable and what is a variable of interest. Exploratory data analysis is fine as long as you report on all tests that you did.

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