Assuming that one works in a scientific discovering frame and is just interested in discovering relations between variables, and not to forecast anything, and since an underspecified regression model leads to biased regression coefficients, I was wondering if it is a statistically sound decision to use the residuals of a fitted model as an additional independent variable to fit again the model in order to get unbiased coefficients of the other independent variables (the "real" independent variables, whose relations with the dependent variable are the object of scientific interest).

  • $\begingroup$ Hi: it's not clear from the question how you know that your model is underspecified ? I've never heard of that approach and my guess is that it wouldn't improve the estimates (since residuals are the left over part of the original regression ) but the relevant question is how you know that the model is under-specified. $\endgroup$
    – mlofton
    Jan 18 '21 at 17:32
  • $\begingroup$ Thanks for the reply @mlofton. I would assume under-specification from evident patterns in residuals, which I would interpret as the presence of an unexplained deterministic element in the model. $\endgroup$
    – N9N9
    Jan 18 '21 at 17:34
  • $\begingroup$ Hi: if you have non-white noise behavior in residuals that would suggest something is being left out or something is mis-specified.I'm not familiar with your idea. Although interesting, you only get the resids after the data comes in, so it's not a practical approach . It's been a while since I considered regression but could you try adding a variable or doing some kind of transformation. Hopefully someone else can reply because I don't feel comfortable giving advice here except that I don't think that using the residuals themselves as an independent variable is not the way to go. $\endgroup$
    – mlofton
    Jan 19 '21 at 18:47
  • $\begingroup$ @mlofton I experimented a bit. Using the residuals does not change the coefficients but the standard errors (smaller) and the significance (higher). Instead, the coefficients can change if you add another variable, so the use of residuals cannot be used in place of a potentially missing variable. Thanks for having discussed the question, it was useful for elaborating further the idea (and finally discard it!) $\endgroup$
    – N9N9
    Jan 19 '21 at 23:35
  • $\begingroup$ Hi: that's interesting though that they decrease the sd. now that you said it, it makes sense that they don't change the coefficient values because, if they did, they would have gotten used in the original step. sometimes experimentation is a great thing. good to meet you. I learned something there also. $\endgroup$
    – mlofton
    Jan 20 '21 at 22:42

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