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I have a standard OLS as follows: Y = intercept + var1 + var2 + var3 + e

I want to run an additional OLS as follows: var3 = intercept + var1 + var2 + e

var1 is the variable of interest; the remaining variables are controls. My theory predicts var1 affects Y and it could also possibly affect var3, hence the second regression.

The question is, would this cause any issues like multicollinearity in the first regression? Any interpretation problems?

Thanks!

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    $\begingroup$ This has been studied considerably in Econometrics. Check Simultaneous Equations Model. It is basically a system of equations. Your model can be run in stages, as long as Y does not enter in the second equation. There should be no problem. $\endgroup$
    – luchonacho
    Aug 26, 2016 at 22:12
  • $\begingroup$ @luchonacho so basically I need to run a two-staged least squares. First get a predicted value for var3 and then use that predicted value to get my estimates for the first regression? $\endgroup$ Aug 26, 2016 at 22:21
  • $\begingroup$ If var3 is data, that is fine. Just run each equation independently. 2SLS is used when you have concerns about endogeneity in one of the regressors, which in your case should not happen. $\endgroup$
    – luchonacho
    Aug 27, 2016 at 7:11
  • $\begingroup$ I think both Vector Auto Regressive models and maybe SEM would allow this. The former uses OLS although commonly it deals with time series data. $\endgroup$
    – user54285
    Aug 26, 2020 at 23:49

1 Answer 1

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Yes -- some important problems, but not intractable ones.

The big issue (and this is easier to see if you draw a graph) is that if your theory is right, you won't get good estimates of the parameters, in the first regression. Really, draw it out -- make a node for every variable, and every time an independent variable is put in a regression, draw an arrow between it, and the variable it's predicting. You'll see that there are two paths from v1, and v2, to Y -- for each, one directly, and one through v3. As a result, the model won't be able to cleanly separate where the influence is coming from, if you throw everything in at once.

What you want to do instead is run the second regression FIRST -- and then, don't use v3, but rather use the residuals from the first regression, i.e. the parts of v3 that v1 and v2 could NOT 'explain.' I.e.

$$ v3 \sim v1 + v2 $$

$$ y \sim v1 + v2 + r3 $$

where $r3$ is $v3 - \hat{v3}$, i.e. what's left over, after prediction. Then (I believe) you'll be okay.

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